UC-NRLF 


B    3    112    Sb7 


IN 


)DS  AND  EXPERIMENTS 
MENTAL  TESTS 


G.A.  RICHARDSON 


METHODS  AND  EXPERIMENTS 
IN  MENTAL  TESTS 


METHODS  AND 
EXPERIMENTS  IN 
MENTAL  TESTS 


BY 

C.  A.   RICHARDSON   M.A.  (Cantab.) 

AUTHOR    OF 
"THE    SIMPLEX    GROUP    INTELLIGENCE    SCALE" 


YONKERS-ON-HUDSON,    NEW    YORK 

WORLD    BOOK    COMPANY 
1922 


WORl-D BOOK     COMPANY 

IHE   HOUSE  OF  APPLIED  KNOWLEDGE 

Established  1905  by  Caspar  W.  Hodgson 
YoNKERs-ON- Hudson,  New  York 
2126    Prairie  Avemte,   Chicago 

Copyright     in     Great     Britain. 
y4ll  rights  reserved 

This  book  is  one  of  a  series  which  will  receive 
additions  from  time  to  time  in  the  way  of  other 
books  written  by  authors  abroad  on  the  general 
subject  of  mental  and  educational  testing.  It  deals 
in  part  with  certain  phases  of  the  technique  of  the 
interpretation  of  test  results  not  generally  appre- 
ciated on  either  side  of  the  water.  All  persons 
who  are  interested  in  testing  will  profit  by  a 
careful  reading  of  the  book. 


Priitied  in  Great  Britain  by  Jarrold  &  Sons,  Ltd.,  Nnrioiek 


PREFACE 

THIS  book  is  an  attempt  to  provide, 
in  a  manner  readily  comprehensible 
to  all  those  interested  in  the  subject, 
an  answer  to  some  of  the  more  important  ques- 
tions that  have  been  asked  as  to  the  nature, 
validity,  and  methods  of  appHcation  of  mental 
tests,  and  the  conclusions  to  be  drawn  there- 
from. It  is  not  an  attempt  to  theorize,  but 
to  illustrate  general  principles  on  a  strictly 
empirical  and  experimental  basis,  and  it  is 
hoped  that  all  who  follow  developments  in 
education  and  in  psychology  may  find  some 
points  of  interest  in  it. 

The  experiment  described  in  the  second  part 
of  Chapter  V  was  carried  out  in  the  county  of 
Northumberland . 

The  note  at  the  end  of  Chapter  IV  was  pub- 
Hshed  in  The  British  Journal  of  Psychology 
(April  1922),  and  I  am  grateful  to  the  Editor 
for  permission  to  include  it  in  this  book. 

5 

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MENTAL  TESTS 

Most  of  the  chapters  of  the  book  were  kindly 
read  by  Mr  Cyril  Burt,  to  whom  I  am  much 
indebted  for  valuable  comments. 

C.  A.  R. 

Newcastle-on-Tyne 
April  1922 


CONTENTS 

CHAPTER  PAGE 

I.   Introductory  9 

II.  The  Reliability  of  the  Stanford- 
BiNET  Scale  of  Intelligence 
Tests  as  an  Index  of  Educable 
Capacity  24 

III.  The  Derivation  of  Mental  Ages 

FROM  Scores  in  a  Group  Test       40 

IV.  Methods  of  Estimating  the  True 

Intelligence       Quotients       of 
Adults  and  Adolescents  56 

V.  The  Reliability  of  the  Group 
Intelligence  Test  as  an  Index 
of  Educability  79 

VI.  Conclusion  89 

Index  93 


Methods  and   Experiments 
in  Mental  Tests 

CHAPTER   I 
INTRODUCTORY 

MENTAL  tests  are  now  becoming  so 
familiar  in  the  English  educational 
world  that  it  is  unnecessary,  in 
introducing  a  book  such  as  this,  to  enter  into 
a  lengthy  description  of  their  nature.  Pre- 
hminary  experimental  work  has  now,  however, 
reached  a  stage  where  we  may  well  pause  to 
look  round  and  to  take  stock  of  the  results  so 
far  reached. 

In  the  first  place,  it  is  necessary  to  consider 
briefly  the  criticisms  which  have  been  levelled 
at  the  tests.  These  criticisms  fall  into  three 
categories  according  as  they  are  directed  at 
(i)  supposed  faults  inherent  in  the  tests  them- 
selves ;  (2)  difficulties  in  carrying  out  the 
alleged  aim  of  the  tests  ;  or  (3)  difficulty  in 
defining  what  this  aim  really  is. 

9 


MENTAL  TESTS 

The  first  kind  of  critic  will  glance  through  a 
test  booklet,  seize  on  some  particular  test,  and 
say  confidently,  "  This  is  far  too  difficult,"  or 
"  This  is  far  too  easy,"  or  "  The  mere  sight  of 
this  will  frighten  the  children  (especially  the 
nervous  ones)  out  of  their  wits,"  or  "  How  can 
you  expect  children  to  understand  long  words 
like  this  ?  " 

The  answer  to  such  a  critic  is  simple  and  con- 
clusive. He  must  be  made  to  understand  that 
the  value  of  a  test  is  not  estimated  by  its  appear- 
ance, but  by  what  it  is  actually  found  to  effect 
in  practice.  That  is,  he  must  be  asked  to  look, 
not  at  the  tests,  but  at  their  results.  The  pro- 
cedure of  mental  testing  is  rigidly  empirical. 
All  tests,  before  any  conclusions  are  drawn  from 
them,  are  tried  on  a  number  of  children  suffi- 
cient to  determine  whether  they  are  suitable  or 
not.  In  the  case  of  a  test  devised  in  this  way 
criticisms  of  the  first  type  are  therefore  met  in 
advance. 

Secondly,  objections  are  sometimes  urged 
against  the  possibility  of  fulfilling  one  primary 
aim  of  the  tests,  namely,  to  give  a  fair  field  and 

10 


INTRODUCTORY 

no  favour.  This  objection  usually  takes  the 
form  of  an  assertion  that  the  tests  can  easily 
be  '  coached  '  for.  It  is  true  that  in  the  present 
individual  scales  (such  as  the  Binet  scale)  a 
few  (not  many)  of  the  tests  are  markedly  sus- 
ceptible to  the  influence  of  coaching,  but  even 
here  all  who  have  had  experience  in  giving  such 
tests  will  know  that  the  presence  of  coaching 
can  be  detected  with  but  little  difficulty.  At 
most  it  is  unlikely  to  add  more  than  a  few  points 
to  the  child's  score.  Moreover,  in  group  testing, 
which  is  the  most  practicable  way  of  carrying 
out  tests  on  a  sufficiently  large  scale,  the  possible 
variety,  not  only  of  the  actual  items  of  the  differ- 
ent tests,  but  also  of  the  particular  forms  which 
those  tests  may  take  (this  being  in  effect  limited 
only  by  the  inventiveness  of  the  human  mind), 
is  so  great  that  coaching  is  rendered  a  waste  of 
time  and  labour.  Even  in  a  standardized  group 
test  which  is  placed  upon  the  market  the  number 
and  variety  of  items  is  sufficiently  large  to 
nullify  the  effects  of  coaching.  In  fact,  the  only 
point  which  can  be  sustained  in  this  criticism 
is  that,  as  children  become  familiar  with  mental 

II 


MENTAL  TESTS 

tests,  the  norms  of  performance  may  slightly 
improve  owing  to  greater  speed  of  getting  to 
work  and  to  absence  of  emotional  disturbance 
due  to  confrontation  with  an  unfamihar  kind  of 
task.  But  any  such  change  in  the  norms  will, 
of  course,  be  automatically  accounted  for  in 
the  interpretation  of  results. 

Thirdly,  it  is  frequently  urged  that  although 
these  tests  are  said  to  be  tests  of  intelligence 
we  are  unable  to  define  what  intelligence  is. 
Now,  in  the  first  place,  it  cannot  be  too  strongly 
emphasized  that  our  ability  to  define  intelli- 
gence is  a  matter  quite  irrelevant  to  the  ques- 
tion of  the  value  of  the  tests.  What  is  wanted  is 
a  method  of  separating  the  sheep  from  the  goats 
— that  is,  the  bright  children,  those  who  are 
capable  of  responding  to  our  efforts  to  educate 
them,  from  the  dull  children,  those  whose 
capacity  for  response  to  educative  influences  is 
distinctly  limited.  If  the  tests  do  in  fact 
enable  us  to  grade  children  in  order  of  their 
educability  it  does  not  matter  in  the  least 
what  we  call  them  ;  though,  since  we  com- 
monly speak   of   bright    children    as    '  intelli- 

12 


INTRODUCTORY 

gent,'  '  intelligence  tests  '  is  probably  eis  good 
a  name  as  any. 

But  apart  from  this  we  can  frame  a  fairly 
precise  definition  of  intelligence  without  much 
difficulty.  In  fact,  it  might  even  be  regarded  as 
synonymous  with  '  educabihty,'  the  two  terms 
being  taken  to  imply  the  ability  to  acquire 
knowledge  and  to  use  it.  Intelligence  is  thus  the 
capacity  to  organize  in  a  coherent  manner  the 
confused  mass  of  experience  which  pours  in 
upon  us  through  the  channels  of  the  senses. 
Or  we  may  even  carry  our  analysis  a  step  farther, 
and  search  for  a  general  factor  which  enters  into 
all  the  processes  which  are  ordinarily  described 
as  '  intelligent.'  To  the  writer  it  seems  that 
attention  constitutes  such  a  factor,  a  belief 
which  Binet  himself  appears  to  have  held.  The 
elementary  intelligent  processes — discrimina- 
tion, comparison,  analysis,  synthesis — are  all 
functions  of  attention.  Intelligence  would 
then  be  defined  as  "  the  functional  efficiency  of 
attention." 

But,  it  may  be  urged,  if  there  is  such  a  general 
factor,  why  is  viot  everybody  equally  good,  or 

13 


MENTAL  TESTS 

equally  bad,  at  everything  ?  Why  should  not 
the  great  mathematician,  for  example,  be  an 
equally  great  historian,  or  zoologist,  or  literary 
critic  ?  The  writer  believes  the  answer  to  be 
that  those  special  aptitudes  with  which  we  are 
so  familiar  in  ordinary  life  are  in  each  case  the 
consequence  of  a  combination  of  general  intelli- 
gence and  special  interest,  the  latter  determining 
the  particular  channels  into  which  the  former  is 
directed.  For  it  is  certainly  a  fact  that  good 
children,  when  young,  are  generally  good  all 
round,  special  tendencies  becoming  markedly 
apparent  only  during  adolescence,  when  all 
kinds  of  interests  and  impulses  hitherto  latent 
begin  to  come  into  action.  Moreover,  a  study  of 
recent  work  in  psycho-analysis  will  reveal  to 
what  a  great  extent  the  direction  in  which  intelli- 
gent activity  turns  is  determined  by  causes  of 
which  the  individual  concerned  is  quite  uncon- 
scious. 

There  is,  however,  another  experimental  result 
at  which  the  critics  of  mental  tests  frequently 
express  incredulous  surprise — namely,  the  fact 
that  intelligence   appears   to  cease  developing 

14 


INTRODUCTORY 

at  something  over  seventeen  years  of  age. 
But  is  this  so  incredible  ?  Surely  not  when 
we  remember  the  nature  of  intelligence.  Intelli- 
gence is  not  knowledge,  but  the  ability  to  organ- 
ize and  employ  knowledge.  Knowledge  may 
grow  indefinitely,  but  the  growth  of  the  ability 
to  deal  with  knowledge  may,  like  physical 
growth,  cease  comparatively  early  in  life. 
The  experienced,  worldly-wise  man  of  forty 
differs  from  the  callow  youth  of  sixteen  or 
eighteen  not  in  ability  to  handle  experience, 
but  in  having  a  far  greater  amount  of  experience 
to  handle,  and  thus  to  apply  in  meeting  new 
situations. 

One  final  word  as  to  criticism  :  the  critics 
should  remember  that  the  results  reached  by 
those  engaged  in  work  on  mental  tests  are  the 
product  of  careful  and  extensive  experimental 
work  carried  out  in  accordance  with  the  usual 
rules  of  scientific  procedure.  Hence  no  criticism 
can  be  effective  unless  it  is  based  on  experiments 
equally  careful  and  extensive. 

We  have  had  occasion  to  draw  a  definite 
distinction  between  intelligence  and  knowledge 

15 


MENTAL  TESTS 
or  attainment.  The  latter  is  a  joint  product 
of  the  former  and  environmental  influences  (of 
which,  of  course,  teaching  is  one  of  the  most 
important).  But  experience  has  shown  that 
intelligence  itself  is  native  to  the  child — a  gift, 
good  or  bad,  at  birth,  which  no  means  known 
to  us  can  appreciably  improve.  The  bright 
child  remains  bright,  the  dull  child  dull,  to  the 
end  of  the  chapter.  We  cannot  increase  intelli- 
gence, but  we  can  enable  the  child  to  make 
the  best  use  of  that  amount  of  intelligence  which 
he  may  happen  to  possess,  and  this  is  the  true 
aim  of  all  education. 

Intelligence  tests,  then,  are  in  no  sense  tests 
of  teaching.  They  differ  from  the  ordinary 
examination  chiefly  in  three  important  ways  : 
(i)  they  are  directed  to  the  estimation  of  the 
child's  natural  gifts,  independently  of  such 
influences  as  home  and  social  environment, 
and  the  effects  of  good  or  bad  teaching ; 
(2)  they  aim  at  enabling  us  to  form  a  sufficiently 
accurate  estimate  of  a  child's  educability  before 
that  child  has,  in  fact,  been  educated  to  any 
great  extent ;  (3)  their  results  are  expressed  in 
16 


INTRODUCTORY 

terms  of  a  purely  objective  standard  and  are 
unaffected  by  any  such  subjective  factor  as  the 
personal  equation  of  a  particular  examiner. 

With  these  aims  in  view  the  tests  are  so 
devised  as  to  presuppose  as  little  as  possible 
beyond  the  fact  that  the  child  tested  has  been 
brought  up  (at  whatever  social  level)  under  the 
ordinary  conditions  of  a  civilized  community, 
and  they  are,  of  course,  graded  so  as  to  be 
applicable  to  children  of  all  ages  and  of  all 
degrees  of  brightness.  Their  ability  to  predict 
(whereas  the  ordinary  examination  is  in  general 
really  effective  only  after  the  event)  should  be, 
if  proven,  of  the  greatest  value  in  enabling  us 
to  determine  for  each  child  the  method  of 
education  fitted  to  make  the  best  of  his  particular 
level  of  intelligence. 

There  are  various  methods  of  expressing 
degree  of  intelligence.  One  of  the  best  known 
is  by  means  of  *  mental  age  '  and  '  intelligence 
quotient '  (I.Q.)-  The  mental  age  of  a  child 
is  the  age  of  the  average  child  to  whom  he  is 
equal  in  intelligence,  and  the  I.Q.  is  the  per- 
centage ratio  of  the  mental  age  to  the  actual 
2  17 


MENTAL  TESTS 

age.  Thus  a  child  of  ten  years  may  have  a 
mental  age  of  twelve  years,  his  LQ.  then  being 
120.  Experiment  has  shown  that  the  LQ. 
of  a  given  child  remains  nearly  constant  (which 
makes  it  a  very  useful  measure),  so  that  the 
child  in  our  example  would,  at  the  age  of  five 
years,  have  had  a  mental  age  of  six  years. 

A  second  way  of  expressing  level  of  intelligence 
is  by  means  of  '  percentile  rank.'  If  a  number 
of  children  be  ordered  according  to  degree  of 
intelligence  the  percentile  rank  (P.R.)  of  any 
one  of  them  is  that  percentage  of  the  whole 
group  which  he  just  exceeds  in  intelligence. 
Thus  if  70  per  cent,  of  the  children  in  the  group 
are  below  the  particular  one  selected  in  intelli- 
gence his  P.R.  will  be  70.  In  view  of  what 
has  been  said  of  the  nature  of  intelligence  it 
is  evident  that  the  P.R.  remains  constant,  for 
if  a  given  child  exceeds  a  certain  percentage  of 
all  children  in  intelligence  he  will  always  exceed 
that  percentage. 

A  third  method,  which  is  often  very  useful  and 
significant,  is  to  express  degree  of  intelligence 
in  terms  of  '  standard  deviation.'  If  a  group 
18 


INTRODUCTORY 

of  children  of  about  the  same  age  be  tested, 
and  the  average  or  mean  of  their  scores  found, 
the  deviation  of  each  child's  score  from  this 
mean  is  obtained  and  squared  ;  the  squares  are 
then  added  and  their  mean  found  by  dividing 
by  the  number  of  children.  The  square  root 
of  this  mean  is  the  standard  deviation  (S.D.). 
It  is  a  measure  of  the '  scatter  '  of  the  intelligence 
of  the  group,  the  intelligence  of  a  given  child  being 
expressed  by  saying  that  his  deviation  from  the 
mean  is  so  many  times  the  standard  deviation. 

Other  useful  quantities  in  connexion  with  the 
scores  of  a  group  of  children  are  the  '  median ' 
— that  is,  the  middle  score  of  the  group  (namely 
that  made  by  the  child  whose  P.R.  is  50) — the 
'  lower  quartile  '  (the  score  made  by  the  child 
of  P.R.  25),  and  the  '  upper  quartile '  (the 
score  made  by  the  child  of  P.R.  75).  Half  the 
difference  between  the  two  quartiles  is  termed 
the  '  semi-interquartile  range '  (S.I.R.),  and 
provides  another  measure  of  scatter  and  one 
easier  to  obtain  than  the  S.D. 

Experiments  in  intelligence  tests  turn  largely 
on  the  comparison  of  the  order  in  which  children 

19 


MENTAL  TESTS 

are  ranged  by  such  tests  with  the  order  in 
which  they  are  ranged  by  other  kinds  of  tests. 
The  results  of  this  comparison  are  expressed 
by  means  of  what  is  termed  a  '  correlation 
coefficient.'  It  would  be  out  of  place  here  to 
enter  into  a  description  of  the  mathematical 
methods  by  which  the  degree  of  correlation  is 
determined.^  It  may  be  stated,  however,  that 
any  value  of  the  correlation  coefficient  numeric- 
ally greater  than  zero  is  presumptive  evidence 
of  the  existence  of  a  real  connexion  between 
the  two  sets  of  quantities  compared  ;  but  for 
small  values  of  the  correlation  coefficient  the 
'  probable  error  '  (P.E.)  is  so  large  as  to  make 
the  result  practically  insignificant.  As  the 
magnitude  of  the  correlation  increases,  however, 
the  probability  of  the  existence  of  a  real  con- 
nexion becomes  rapidly  greater,  and  when  the 
value  is  about  .5  the  probability  of  some 
connexion  begins  to  approach  reasonable  cer- 
tainty. For  values  of  the  order  .9  and  upward 
the    existence    of    a   very    close    connexion    is 

^   For  such   a  description  see,  e.g.,   Brown    and   Thomson,    The 
Essentials  of  Mental  Measurement.  Part  II. 

20 


INTRODUCTORY 

practically  certain.  Perfect  correlation  is  repre- 
sented by  the  value  i.oo. 

In  the  descriptions  of  the  experiments  which 
follow  the  writer  has  been  concerned  primarily 
to  illustrate  the  general  principles  involved  in 
the  examination  of  the  validity  of  intelligence 
tests,  and  (if  that  validity  is  established)  in  the 
application  of  the  tests  for  educational  purposes. 
He  has  accordingly  refrained  from  explaining  in 
detail  the  more  technical  mathematical  and 
statistical  processes  employed  in  obtaining  some 
of  the  results. 1 

In  actually  carrying  out  the  experiments  the 
writer  had  two  chief  aims  in  view.  In  the  first 
place,  he  wished  to  make  the  investigation  not 
merely  extensive,  but  also  intensive.  In  other 
words,  he  was  not  concerned  simply  to  get  mass 
results,  but  also  to  make  a  close  study  of 
individual  children.  For  it  is  clear  that  we  can 
never  be  satisfied  with  the  tests  until  we  are 
confident  that  they  are  not  only  reliable  for 
groups  of  children  considered  as  wholes,   but 

'    Such  explanations  may  be  found  elsewhere  ;  e.g.,  in  Brown  and 
Thomson,  op.  cit.,  Part  II. 

21 


MENTAL  TESTS 

are  also  reasonably  certain  to  do  justice  to 
particular  children.  We  require  them  to  be 
at  least  as  certain  in  this  respect  as  the  older 
methods  of  selection. 

Secondly,  the  writer  has  relied  as  far  as 
possible  only  on  objective  standards  of  com- 
parison. In  judging  intelligence  tests  a  method 
frequently  employed  is  to  compare  their  results 
with  teachers'  estimates.  Now  while  no  one 
would  doubt  the  value  of  the  estimate  of  a 
competent  teacher,  owing  to  the  fallibility  of 
human  nature  we  can  never  feel  quite  the 
same  certainty  when  a  subjective  factor,  such 
as  the  personal  equation  of  the  teacher,  enters 
into  our  calculations  as  we  can  when  the 
standard  of  reference  is  purely  objective.  A 
man  may  be  trained  to  estimate  distances  with 
some  accuracy  merely  from  observation,  but 
we  should  never  trust  his  estimate  as  we  trust 
the  foot-rule.  In  this  connexion,  therefore, 
it  is  necessary  to  state  only  that  the  writer, 
like  others,  has  found  close  general  agree- 
ment between  teachers'  estimates  and  results 
of  the  tests. 

22 


INTRODUCTORY 

The  chief  problems  attacked  in  the  experi- 
ments to  be  described  were  the  reUabihty  of 
the  individual  test  and  of  the  group  test  as 
indexes  of  educabihty,  and  the  methods  of 
interpreting  scores  in  group  tests  in  such  a  way 
as  to  form  an  estimate  of  the  mental  capacities 
of  the  subjects  tested — in  particular,  the  methods 
to  be  adopted  in  the  difficult  case  of  adults  and 
adolescents. 


23 


CHAPTER  II 

THE  RELIABILITY  OF  THE  STANFORD-BINET 
SCALE  OF  INTELLIGENCE  TESTS  AS  AN 
INDEX  OF  EDUCABLE  CAPACITY 

INTRODUCTION.  The  problem  of  deter- 
mining the  use  that  can  be  made  in 
educational  work  of  intelligence  tests  in 
general,  and  of  the  Stanford-Binet  scale  in 
particular,  reduces,  as  we  have  seen,  essentially 
to  this  :  Is  it  possible  to  foretell  by  means  of 
these  tests  the  limitations  of  a  child's  capacity 
for  being  educated,  and  hence  to  lay  down  the 
appropriate  lines  along  which  he  should  be 
taught  ? 

The  experiment  herein  described  was  an 
attempt  to  obtain  decisive  evidence  of  a  quanti- 
tative nature  on  this  point.  Evidently  the  most 
direct  method  to  adopt  was  to  range  a  number 
of  children  in  order  of  intelligence  as  determined 
by  the  S.B.  scale,  and  then  to  arrange  them  in 
order  of  educable  capacity  by  some  entirely 
different  (though  sufficiently  accurate)  means 
24 


THE   STANFORD-BINET  SCALE 

of  estimation.  The  two  orders  could  then  be 
compared,  and  the  correlation  between  them 
calculated. 

For  this  purpose  five  batches  of  children 
(about  twenty  in  each  batch)  were  taken  from 
five  different  schools.  In  order  to  ensure  that 
the  children  had  for  some  time  had  sufficiently 
favourable  opportunities  of  learning,  the  schools 
selected  were  those  known  to  be  comparatively 
efficient,  and  the  age  range  of  the  children  was 
from  ten  to  twelve,  so  that  they  had  been  at 
school  for  some  years.  In  these  circumstances 
the  children's  educable  capacity  could  evidently 
be  estimated  by  discovering  what  they  had 
actually  shown  themselves  capable  of  learning. 
As  indexes  of  this  capability  arithmetic  and 
English  composition  were  chosen  as  being  the 
most  fundamental,  as  well  as  the  most  repre- 
sentative, subjects  of  the  curriculum. 

The  children  included  members  of  both  sexes, 
and  they  ranged  in  intelligence  from  exceptional 
superiority  to  definite  mental  defectiveness. 
In  order  to  discount  the  effect  of  such  hetero- 
geneities as  difference  of  sex,  different  methods 

25 


MENTAL  TESTS 

of  instruction,  etc.,  each  batch  was  considered 
separately,  in  addition  to  the  consideration  of 
the  group  as  a  whole. 

Every  child  was  tested  individually  by  the 
S.B.  scale,  and  an  order  drawn  up  from  the 
results. 

In  estimating  the  significance  of  what  follows 
it  should  be  borne  in  mind  that  extensive  ex- 
periments in  America — in  the  course  of  which 
children  were  retested  under  varying  conditions, 
by  different  examiners,  and  at  varying  intervals 
— have  shown  that  the  intelligence  quotient  of 
a  given  child  remains  practically  constant. 
Indeed,  the  average  variation  of  I.Q.,  even 
over  intervals  of  several  years,  is  no  more  than 
about  5  per  cent.^  Hence,  had  the  children  in 
the  experiment  to  be  described  been  tested  when 
entering,  say,  the  senior  school — i.e.,  at  about 
seven  years  of  age — instead  of  at  about  eleven 
years  of  age,  the  order  obtained  would  have 
been  practically  the  same. 

After  being   tested   by   the   S.B.    scale    the 

1  Cf.,  e.^.,  Terman,  The  Intelliqence  of  School  Children  (Harrap), 
Chapter  IX,  especially  pp.  138  ff. 

26 


THE   STANFORD-BINET  SCALE 

children  were  set  a  paper  in  applied  arithmetic. 
For  this  purpose  the  paper  given  on  pp.  193  ff. 
of  Dr  Ballard's  Mental  Tests  was  used.  On 
the  child's  performance  in  this  test  it  was 
possible  to  assign  to  him  an  '  arithmetical ' 
mental  age,  and  hence  to  find  his  '  arithmetic 
quotient '  (A.Q.)  by  a  process  similar  to  the 
calculation  of  the  I.Q. 

At  this  point  it  should  be  noted  that  the 
norms  of  performance  taken  differed  somewhat 
from  those  given  on  p.  195  of  Mental  Tests. 
It  became  apparent  early  in  the  experiment 
that  these  were  too  low,  and  in  fact  Dr  Ballard 
mentions  the  possibiUty  of  this  on  p.  195  of  his 
book.  The  norms  he  gives  are  in  arithmetical 
progression,  and  it  therefore  seemed  probable 
that  any  increase  due  to  a  return  of  more 
favourable  conditions  of  education  would  be 
marked  by  a  multiplication  of  his  norms  by  a 
constant  factor  (i.e.,  the  same  for  each  norm 
at  a  given  time).  The  first  batch  of  results 
indicated  an  increase  of  about  25  per  cent,  in 
the  norms,  and  this  was  strikingly  borne  out 
as  further  results  came  in. 

27 


MENTAL  TESTS 

In  composition  the  children  were  required  to 
write  on  one  of  the  following  subjects  : 
(i)  How  I  spent  my  Easter  Holiday. 

(2)  Describe  what  you  have  seen  of  the 
River  X  and  its  banks. 

(3)  Write  a  story  called  Look  before  you  Leap. 

(4)  Write  about  this  piece  of  poetry  : 

Bright  yellow,  red,  and  orange, 
The  leaves  come  down  in  hosts  ; 
The  trees  are  Indian  princes, 
But  soon  they'll  turn  to  ghosts. 

As  was  expected,  about  75  per  cent,  of  the 
children  chose  the  first  subject,  but  this  did 
not  matter,  as  little  difficulty  was  found  in 
differentiating  them. 

In  order  to  obtain  an  accurate  standard  a 
panel  of  five  experienced  examiners  was  formed, 
and  every  composition  was  marked  indepen- 
dently by  each  examiner.  A  special  method  of 
marking  was  arranged.  The  examiners  did  not 
know  the  age  of  any  of  the  children,  and  they 
were  asked  to  mark  a  composition  by  assigning 
the  child  an  age  on  it  ;  that  is,  a  judgment  of 
this  kind  was  formed  :  "  This  composition  is 
28 


THE  STANFORD-BINET  SCALE 

about  equal  to  that  of  the  average  child  of 
%  years  of  age."  The  mark  %  was  then  awarded. 
The  mean  of  the  five  markings  was  found,  and 
this  was  taken  as  the  child's  mental  age  as 
regards  attainment  in  Enghsh  composition.  In 
awarding  marks  the  examiners  were  asked  to 
give  primary  regard  to  power  of  thought  and 
to  ability  to  express  and  arrange  ideas  in  a 
logical  and  coherent  manner.  It  may  be 
remarked  that,  in  the  majority  of  cases, 
there  was  fairly  close  agreement  between  the 
examiners. 

From  the  mental  age  thus  obtained  the 
'  composition  quotient '  (C.Q.)  was  calculated. 

An  order  was  drawn  up  by  taking  the  mean 
of  the  two  mental  ages  (for  arithmetic  and 
composition  respectively)  for  each  child,  and 
another  by  taking  the  mean  of  the  A.Q.  and 
C.Q.  (giving  what  may  be  called  the  '  educabihty 
quotient' — E.Q.).  Evidently  this  last  order 
is  the  significant  one  from  the  point  of  view  of 
educabihty. 

Results  of  the  Experiment.  In  working 
out  the  correlations  for  the  separate  batches 

29 


MENTAL  TESTS 

Pearson's  correction  of  Spearman's  formula  for 
ranks  was  employed  (this  was  sufficiently  accu- 
rate for  correlations  of  the  order  obtained). 

Calculated  from  the  quotients  the  results  were 
as  follows  : 


Group   Group    Group    Group   Group 
A  B  C  D  E 


Correlation    between 
educability  and  gene- 
ral     intelligence      as     "^(Boys)      (Girls)    (Boys)      (Girls)     (Boys) 
measured      by     the    1    .93         .75        .88        .92         .91 
S.B.  scale  ) 

Calculated  from  the  mental  ages  the  corre- 
lations were  : 

Group  A  Group  B  Group  C  Group  D  Group  E 
•93     77     -89     -93     -92 

It  will  be  seen  that  these  correlations  are 
exceptionally  high.^  Moreover,  the  deviations 
from  perfect  correlations  are  almost  entirely 
due  to  not  more  than  about  15  per  cent,  of  the 
children.     For  example,  in  Group  B,  where  the 


^  It  must,  of  course,  be  remembered  that  the  high  magnitude  of 
the  correlations  here  and  in  the  sequel  is  in  part  due  to  the  wide 
range  of  ability  of  the  children  tested,  although  this  does  not  in 
any  way  vitiate  the  result.  In  the  case  of  a  group  of  children  of 
approximately  the  same  ability,  such  as  a  fairly  uniform  class,  this 
homogeneity  would  alone  tend  to  reduce  the  correlation,  apart  from 
any  other  factor. 

30 


THE  STANFORD-BINET  SCALE 

correlations  are  lowest,  the  deviation  was  due 
very  largely  to  three  of  the  children,  apart  from 
whom  the  correlation  coefficient  would  have 
been  about  .9.  In  these  cases  there  were  almost 
invariably  obvious  explanations  of  the  dis- 
crepancy. Thus  many  of  the  cases  were  those 
of  young,  bright  children  who  had  not  been 
given  sufficient  promotion,  and  therefore  had 
had  no  opportunity  of  reaching  a  level  of 
attainment  appropriate  to  their  mental  age. 
Other  causes  of  fall  of  E.O.  below  I.Q.  were 
late  entry,  bad  attendance,  inferior  home  con- 
ditions, serious  emotional  disturbance,  physical 
defects,  and  malnutrition.  Cases  of  marked 
rise  of  E.Q.  above  1.0.  are  rare  (amounting  in 
this  experiment  to  only  about  4  per  cent,  of 
the  whole)  and  seem  to  be  generahy  attributable 
to  a  combination  of  mediocre  or  inferior  inteUi- 
gence  with  some  temperamental  characteristic 
such  as  capacity  for  hard  work,  or  with  home, 
school,  or  social  conditions  distinctly  above  the 
average.  In  such  cases,  however,  it  is  probable 
that  the  superiority  of  E.Q.  to  I.Q.  is  not 
maintained  to  the  same  extent  as  the  children 

31 


MENTAL  TESTS 

grow  older.  At  all  events,  it  is  clear  that 
the  few  discrepancies  noted,  far  from  detract- 
ing from  the  reliability  of  the  S.B.  scale, 
rather  went  to  confirm  its  value  when  properly 
used. 

At  the  other  end  of  the  scale  converse  results 
add  further  confirmation.  For  of  the  children 
of  markedly  inferior  intelligence  every  one  was 
placed  in  the  attainment  tests  in  practically 
exactly  the  position  assigned  to  him  by  the 
S.B.  scale.  Hence  the  verdict  of  the  latter — 
namely,  that  these  children,  even  in  favourable 
circumstances,  were  incapable  of  learning  much 
— was  completely  vindicated. 

One  particular  point  is  of  interest  in  the  last 
connexion.  The  norm  of  performance  in  the 
arithmetic  test  at  eight  years  old  is  f  of  a 
mark.  Not  a  single  child  whose  mental  age, 
as  given  by  the  S.B.  scale,  was  below  eight  years 
scored  a  mark  in  the  arithmetic  paper.  The 
significance  of  this  becomes  clear  in  the  light 
of  the  facts  that  (i)  some  of  these  children 
were  actually  twelve  years  old ;  and  (2)  the 
first  problem  in  the  arithmetic  paper  was  as 
32 


THE  STANFORD-BINET  SCALE 

follows  :  "If  there  are  lOO  apples  on  a  tree  and 
the  wind  blows  down  17,  how  many  are  left  on 
the  tree  ?  "  Some  of  the  children  diagnosed 
by  the  S.B.  scale  as  mentally  defective  wrote 
pages  of  figures,  but  without  scoring  a  single 
mark. 

The  group  of  100  children  was  then  considered 
as  a  whole.  The  rank  correlation  between  the 
results  of  the  educational  tests  and  those  of  the 
S.B.  test  was  obtained  for  the  whole  group. 
Calculated  from  the  mental  ages  the  correlation 
was  .90,  and  from  the  quotients  .89.  The 
absolute  (as  distinguished  from  the  '  rank ') 
correlation  between  the  I.Q.s  and  E.Q.s  of  the 
members  of  the  group  was  also  calculated  by 
the  Bravais-Pearson  product-moment  formula, 
and  was  found  to  be  .89.  The  probable 
error  in  these  three  correlations  was  of  the 
order  .01. 

It  will  be  seen  from  the  table  at  the 
end  of  this  chapter  that  the  correspondence 
between  I.Q.  and  E.Q.  was,  in  the  majority 
of  cases,  strikingly  close.  For  the  whole 
group  the  distributions  of  both  I.Q.  and  E.Q. 
3  33 


MENTAL  TESTS 

approximated    to    normal.     The    constants  of 
these  distributions  were  : 

I.Q.  E.Q. 

Highest  149  Highest  138 

Lowest  61  Lowest  64 

Mean  104.36  Mean  102.33 

Median  102  Median  102 

Semi-interquartile  Semi-interquartile 

range  13  range  9.5 

Standard  deviation  Standard  deviation 

19.13  14.56 

It  will  be  seen  that  the  scatter  of  the 
E.Q.s  is  markedly  less  than  that  of  the  I.Q.s, 
probably  indicating  insufficient  elasticity  of 
promotion  so  far  as  the  brighter  children  are 
concerned. 

Lastly,  it  is  important  to  remember  that,  for 
reasons  already  pointed  out,  the  S.B.  scale 
would  have  arranged  these  children  in  practi- 
cally the  same  order  had  they  been  tested  at 
(say)  about  seven  years  old  instead  of  at  about 
eleven.  In  other  words,  the  order  of  intelligence 
given  by  the  S.B.  scale  at  about  seven  years  old 
would  correlate  with  the  order  of  the  children's 
educational  attainment  at  about  eleven  years  old 
34 


THE  STANFORD-BINET  SCALE 

to  nearly  the  same  degree  as  the  correlations 
given  above.  Hence,  by  testing  at  seven  years 
old  it  would  be  possible  accurately  to  predict 
educational  attainments  after  some  years' 
interval — that  is,  in  effect,  to  form  an  exact 
estimate  of  educable  capacity. 

Conclusions.  Here,  then,  we  have  a  num- 
ber of  children  of  both  sexes,  educated  in 
different  schools,  and  of  all  levels  of  intelligence, 
and  it  turns  out  that  their  general  intelligence, 
as  assigned  by  the  S.B.  scale,  correlates  to  a 
high  order  with  their  educable  capacity  as 
estimated  from  what  they  have  actually  shown 
themselves  capable  of  learning  in  favourable 
circumstances.  In  about  85  per  cent,  of  the 
cases  the  correlation  is  practically  perfect,  while 
in  the  remaining  15  per  cent,  there  are  generally 
clear  reasons  for  the  discrepancy. 

It  is  true  that,  for  many  purposes,  the  number 
of  children  involved  would  not  be  sufficient  to 
warrant  the  induction  of  a  general  principle  ; 
but  it  should  be  pointed  out  that  the  circum- 
stances were  such  as  to  render  the  quantitative 
evidence  afforded  by  this  experiment  practically 

35 


MENTAL  TESTS 

decisive  in  favour  of  the  reliability  of  the  S.B. 
scale  as  an  index  of  educable  capacity,  especially 
when  this  evidence  is  taken  in  conjunction  with 
the  mass  of  quahtative  evidence  accumulated 
in  America  by  comparison  with  teachers'  esti- 
mates, following  up  of  school  careers,  etc.  For 
there  does  not  appear  to  be  any  reason  why 
this  group  of  children  should  differ  markedly 
from  other  groups  of  children,  educated  under 
sufficiently  favourable  conditions,  in  such  a  way 
as  to  modify  the  results  of  the  experiment 
materially.  In  other  words,  there  seems  no 
reason  to  doubt  that  this  group  may  be  taken 
as  a  typical  example  of  groups  of  children  who 
have  been  afforded  satisfactory  opportunities 
of  learning  ;  and,  if  this  be  granted,  it  evidently 
follows  that  conclusions  true  of  this  group  would 
also  be  true,  in  their  main  principles,  of  any 
similar  group. 

There  are  also  certain  subsidiary  conclusions 
of  interest.  Looking  at  the  matter  from  the 
converse  point  of  view,  it  follows  that  tests  in 
arithmetic  and  English  composition  are  good 
indexes  of  educable  capacity,  always  provided  that 

36 


THE  STANFORD-BINET  SCALE 

(i)  the  tests  are  properly  standardized  ;  (2)  the 
children  to  whom  they  are  applied  have  been 
educated  in  favourable  circumstances ;  and 
(3)  due  allowance  is  made  for  the  ages  of  the 
individual  children. 

Attainment  in  arithmetic  seems  to  correlate 
with  general  inteUigence  rather  more  closely 
than  does  attainment  in  composition,  for  whereas 
the  respective  correlations  for  the  five  groups  of 
children  between  general  intelligence  and  arith- 
metical attainment  were  .91,  .78,  .87,  .92,  .88 
(calculated  from  the  quotients),  or  .92,  .83,  .90, 
.93,  .86  (calculated  from  the  mental  ages),  the 
correlations  between  general  intelligence  and 
attainment  in  composition  were  .93,  .42,  .72, 
.84,  .70  (calculated  from  the  quotients),  or 
.89,  .32,  .80,  .82,  .82  (calculated  from  the  mental 
ages).  Probably  the  lower  correlation  in  the 
case  of  composition  is  partly  real,  partly  appar- 
ent. Partly  apparent,  for,  even  with  five 
independent  markings,  it  is  not  possible  to 
obtain  an  assessment  of  ability  in  composition 
of  the  same  accuracy  as  the  assessment  of 
arithmetical  abihty.     Partly  real,  for  it  seems 

37 


MENTAL  TESTS 

probable  that  attainment  in  composition  depends 
on  other  factors  (such  as  home  and  social 
environment)  besides  intelligence  and  education 
to  a  greater  extent  than  does  attainment  in 
arithmetic.  The  reason  for  the  comparative 
breakdown  of  the  correlation  as  regards  com- 
position in  Group  B  is  not  altogether  clear. 

If  the  S.B.  scale  is  accepted  as  a  reliable  index 
of  educable  capacity  children  might  well  be 
tested  by  it  twice  during  their  school  hfe,  namely 
at  about  seven  years  of  age,  and  again  at  about 
eleven  years  of  age.^  The  first  test  would 
indicate  the  child's  educabihty  with  sufficient 
accuracy  to  determine  the  methods  according  to 
which  he  should  be  taught  in  the  senior  school. 
The  second  test  would  serve  to  check  the  I.Q. 
obtained  from  the  result  of  the  first  test,  and 
would  also  settle  the  question  of  the  child's 
capability  of  profiting  by  advanced  instruction. 

^  For  children  of  eleven,  however,  it  will  probably  be  necessary 
to  use  group  intelligence  tests,  owing  to  the  immense  saving  of 
time  thereby  effected. 


38 


Child's 

No. 

I.Q. 

E.Q. 

Child's 

No. 

I.Q. 

E.Q. 

.Child's 

,  No. 

I.Q. 

E.Q. 

I 

67 

71 

37 

91 

104 

72 

96 

98 

2 

141 

138 

38 

112 

113 

1Z 

lOI 

100 

3 

lOI 

107 

39 

98 

107 

74 

96 

100 

4 

126 

124 

40 

95 

91 

75 

122 

112 

5 

1Z 

69 

41 

85 

88 

76 

no 

105 

6 

132 

125 

42 

III 

114 

78 

95 

97 

7 

61 

64 

43 

142 

130 

i  79 

102 

107 

8 

112 

100 

44 

99 

89 

80 

lOI 

102 

9 

109 

105 

45 

92 

91 

81 

102 

97 

10 

119 

98 

46 

116 

108 

82 

89 

88 

II 

125 

108 

47 

83 

81 

83 

89 

92 

12 

81 

100 

48 

114 

106 

85 

131 

118 

13 

123 

113 

49 

90 

92 

86 

115 

no 

14 

78 

81 

50 

91 

84 

87 

71 

82 

15 

121 

112 

51 

80 

83 

88 

138 

130 

i6 

75 

67 

52 

88 

87 

89 

128 

120 

17 

122 

120 

53 

96 

89 

90 

121 

112 

i8 

102 

97 

54 

92 

96 

91 

128 

120 

19 

117 

III 

55 

133 

107 

92 

129 

119 

20 

139 

125 

56 

112 

106 

93 

102 

107 

21 

108 

114 

57 

120 

no 

94 

102 

115 

22 

107 

100 

59 

III 

121 

95 

105 

104 

23 

71 

75 

60 

130 

108 

96 

107 

108 

24 

113 

108 

61 

92 

105 

97 

107 

100 

27 

92 

lOI 

62 

86 

94 

98 

115 

no 

28 

144 

125 

63 

no 

102 

99 

108 

103 

29 

149 

117 

64 

113 

117 

100 

88 

95 

30 

108 

99 

65 

102 

107 

lOI 

96 

82 

31 

103 

lOI 

66 

67 

76 

102 

78 

98 

32 

92 

92 

67 

89 

95 

103 

91 

93 

33 

93 

90 

68 

128 

121 

104 

99 

98 

34 

113 

102 

69 

84 

94 

105 

94 

94 

35 

82 

102 

70 

78 

92 

36 

117 

114 

71 

134 

134 

The  missing  numbers  are  those  of  children  who  were 
absent  from  some  of  the  tests. 

39 


CHAPTER  III 

THE  DERIVATION  OF  MENTAL  AGES   FROM 
SCORES  IN  A  GROUP  TEST 

IF  a  number  of  children  have  been  examined 
by  means  of  a  group  inteUigence  test  we  are 
able,  by  comparing  their  scores,  to  arrange 
them  in  order  of  relative  intelligence.  This  is 
sufficient  for  the  important  purpose  of  classifying 
the  children  according  to  the  degree  of  develop- 
ment which  their  intelligence  has  reached.  But 
it  is  quite  inadequate  for  other,  and  equally 
important,  purposes.  For  we  cannot  compare 
the  real  degrees  of  intelligence  of  the  children 
until  we  have  taken  their  ages  into  account. 
For  example,  one  child  may  score  more  than 
another  in  the  test,  but  the  first  child  may  be 
fourteen  years  of  age,  while  the  second  may 
be  only  ten.  Thus,  although  the  intelligence 
of  the  first  has  developed  more  than  that  of  the 
second  (so  that  he  would  be  classified  higher), 
the  latter  may  in  relative  development  be  ahead 
of  the  former.  That  is,  the  ten-year-old  may  be 
40 


DERIVATION  OF  MENTAL  AGES 

more  developed  than  was  the  fourteen-year-old 
at  ten  years  of  age.  Therefore,  before  we  can 
be  in  a  position  to  judge  as  to  which  child  is 
really  the  more  intelligent  we  must  somehow 
manage  to  reduce  them  to  a  common  denomi- 
nator by  taking  their  ages  into  account. 

The  process  we  have  indicated  is  carried  out 
(as  we  have  seen  in  another  connexion)  by 
assigning  to  each  child  a  mental  age,  dividing 
this  by  the  actual  age,  and  multiplying  by  loo 
to  obtain  the  I.Q.  The  question  then  arises  as 
to  how  mental  age  is  to  be  obtained  from  score 
in  the  test.  The  mental  age  is  the  age  of  the 
'  average  '  child  to  whom  the  given  child  is  equal 
in  intelligence.  It  will  therefore  be  necessary 
to  determine  the  score  of  average  children  in 
the  test,  or,  to  put  it  another  way,  to  calculate 
the  average  scores  in  the  test  of  children  in 
various  age  groups.  Thus  we  may  take  all  the 
ten-year-olds  tested,  and  find  their  average 
score  ;  similarly  with  the  eleven-year-olds,  and 
so  forth.  But  if  our  results  are  to  be  vahd  the 
age  groups  must  be  sufficiently  large  and  suffi- 
ciently random  to  constitute  fair  samples. 

41 


MENTAL  TESTS 

Having  obtained  these  average  scores  or 
'  age  norms  '  we  proceed  to  find  the  mental  age 
of  any  given  child  as  in  the  following  example  : 
Suppose  a  child  of  ten  scores  60,  and  we  find 
that  60  is  the  average  score  of  the  twelve-year- 
olds.  Then  the  child's  mental  age,  as  measured 
by  the  test,  is  twelve,  and  his  I.Q.  is  120. 

But  is  the  mental  age,  as  measured  by  the  test, 
the  true  mental  age,  i.e.,  does  it  express  with 
sufficient  accuracy  the  child's  standing  relatively 
to  the  average  child  ?  For  the  aspects  of 
intelhgence  are  so  many  and  various  that  it  is 
not  easy  to  devise  a  test  to  measure  them  all, 
and  it  is  possible  that  we  may  miss  some 
point  in  which  the  child  shows  up  particularly 
well. 

Now  unfortunately  it  is  possible  to  assign  a 
mental  age  only  by  means  of  some  test,  and  no 
test  is  perfectly  comprehensive.  What  is  needed, 
then,  is  a  specially  comprehensive  test  which, 
though  not  perfect,  will  serve  as  a  standard  by 
which  the  results  of  other  tests  may  be  judged. 
Probably  the  most  accurate  and  comprehensive 
scale  of  tests  at  present  available,  at  any  rate 
42 


DERIVATION  OF  MENTAL  AGES 

for  children  up  to  about  fourteen  years  of  age, 
is  the  Stanford-Binet  scale.  It  Vv^ill  therefore 
be  important,  after  deriving  mental  ages  from 
the  group  test  in  the  way  indicated  above,  to 
check  these  mental  ages  by  comparison  with 
the  corresponding  Stanford-Binet  mental  ages 
in  the  case  of  children  who  have  been  tested 
both  by  the  group  test  and  by  the  Stanford-Binet 
scale. 

We  may  now  proceed  to  illustrate  the 
principles  we  have  laid  down  by  an  example 
from  actual  practice. 

472  children,  ranging  from  ten  to  thirteen 
years  of  age,  were  tested  by  Form  A  of  the 
Terman  Group  Test  of  Mental  Ability,  for  which 
no  age  norms  of  performance  had  at  that  time 
been  pubhshed.  After  the  examination  the 
booklets  were  scored  and  the  average  scores  or 
age  norms  for  different  age  groups  calculated. 
A  '  curve  of  performance  '  was  then  obtained 
by  plotting  average  scores  for  various  ages 
against  those  ages.^     Curves  of  performance  in 

1  This  procedure  will  give  valid  results  only  if  the  distribution 
of  scores  in  each  age  group  is  approximately  normal — as  it  actually 
was  in  the  experiment  here  described. 

43 


MENTAL  TESTS 

group  tests  of  this  kind  usually  approximate, 
over  a  considerable  range,  to  straight  lines,  ^ 
and  the  curve  obtained  in  this  case  was  no 
exception.     The  age  norms  found  were  : 


Age  in  years 

loi 

I0| 

Hi 

III 

I2i 

I2| 

I3i 

Average  score 

48 

56 

65 

71 

82 

83 

88 

isjs     i3o    140     190     160     iro 
in  Months 


/^ 


Cy) 


Fig.  I 


Hence  the  curve  of  performance  was  as  in 
Fig.  I. 

1  The  true  curve  of  performance  is  probably  an  ogive  {cf.  foot- 
note on  p.  65).  The  straight  Hnes  obtained  as  described  in  this 
chapter  are  simply  intended  to  be  close  approximations  to  a  large 
portion  of  the  ogive.  The  simple  nature  of  the  equation  to  the 
straight  line  renders  it  very  convenient  to  use  in  practice. 

44 


DERIVATION  OF  MENTAL  AGES 
The  section  AB  is  a  straight  Une,  the  points 
corresponding  to  ages  loj,  lof,  iij,  12^  years 
respectively  lying  practically  exactly  on   this 
line,  while  the  point  corresponding  to  iif  years 
is  very  nearly  on  it.     As  to  the  eccentric  section 
BC  a  word  of  explanation  is  necessary.     After 
the  age  of  about  12  years  many  of  the  brighter 
children  have  left  the  elementary  schools  for 
secondary  and  other  schools.     Hence  the  age 
groups  above  12  become  less  and  less  representa- 
tive samples,  and  their  performances  will  drop 
below  what  might  have  been  expected  from  a 
consideration  of  the  performances  of  the  younger 
children.     We  must  therefore  neglect  the  portion 
BC  of  the  curve  of  performance,  and  replace  it 
by  the  dotted  line  BK,  which  is  an  extension 
of  the  straight  line  AB.     The  higher  mental 
ages  should  then  be  obtained,  not  from  BC, 
but  from  BK,  which  represents  the  average  scores 
of  higher  age  groups  on  the  assumption  that 
intelligence  continues  to  develop  at  the  same 
rate  as  for  the  younger  age  groups.^ 

If  we  call  the  mental  age  y  and  the  score  in 

*  This  point  will  be  elucidated  more  fully  in  Chapter  IV. 

45 


MENTAL  TESTS 

the   group   test   x,    the   straight   Hne   ABK   is 
represented  by  an  equation  of  the  form 
y  =  mx  +  c 

which  is  the  typical  equation  for  a  straight  hne. 

For  ABK  it  can  easily  be  shown  that,  if  y  be 

expressed  in  months,  m  =  ^^  and  c  =  90.     Hence 

y  =  TO  X  +  90 
that  is,  the  mental  age  can  be  derived  from  the 
score  by  the  following  formula  : 

Mental  age  in  months  =  {^^  of  score)  +  90 
Of  course,  once  having  obtained  the  straight 
line  ABK,  it  would  be  possible  to  read  off  from 
it  directly  the  mental  age  corresponding  to  any 
given  score  without  working  out  the  formula. 
Also  a  table  might  be  drawn  up  from  the  line 
or  from  the  formula  giving  the  mental  age  for 
various  scores.  Had  the  curve  of  performance 
not  been  a  straight  line  its  equation  would  have 
been  more  complicated,  and  the  determination 
of  the  formula  might  have  presented  some  diffi- 
culty. In  such  a  case  it  would  be  more  convenient 
to  construct  a  table  direct  from  the  curve. 
So  far  so  good — but  a  fresh  point  now  arose. 

46 


DERIVATION  OF  MENTAL  AGES 

It  was  known  that  the  batch  of  children  tested 
was  Hkely  to  be,  on  the  whole,  above  the 
average  of  the  total  population.  Hence  this 
group  of  children  set  a  standard  above  the 
average,  so  that  the  mental  ages  given  by  the 
above  formula  would  be  lower  than  the  true 
mental  ages  {i.e.,  the  mental  ages  obtained  rela- 
tively to  a  fair  sample  of  the  zd^hole  population, 
for  such  a  sample  w^ould  set  a  lower  standard 
than  the  particular  group  of  children  tested). 
Therefore  the  formula  could  be  regarded  only 
as  a  first  approximation  to  the  truth,  good 
enough  for  arranging  these  particular  children  in 
their  order  of  intelligence,  but  needing  to  be 
checked  and  modified  where  necessary  if  it  was 
to  be  used  for  obtaining  true  mental  age. 

The  formula  was  checked  in  the  following 
way  :  One  hundred  children  were  tested  indivi- 
dually by  the  Stanford-Binet  scale,  and  were 
then  given  the  Terman  group  test.  A  curve  was 
plotted  giving  the  relation  between  score  in  the 
group  test  and  S.B.  mental  age.  As  was 
expected,  this  curve  gave  somewhat  higher 
mental  ages  for  given  scores  than  did  the  line 

47 


MENTAL  TESTS 

ABK  in  Fig.  i.  The  number  of  high  scores 
was  too  few  to  form  a  basis  for  safe  generalization, 
but  in  the  higher  ranges  there  was  an  apparent 
tendency  for  the  new  curve  and  the  Une  ABK 
to  converge  toward  one  another.  This  looked 
as  if  the  formula  obtained  from  ABK  was  more 
accurate  for  high  scores  than  for  low. 

Another  check  became  available  later.  In  a 
more  recent  edition  of  the  Manual  of  Directions 
for  the  Terman  group  test^  there  appeared  a 
table,  based  on  306  cases,  giving  tentatively  the 
probable  correspondence  between  scores  and 
S.B.  mental  ages.  The  line  plotted  from  this 
table  was  found  to  agree  very  closely  indeed 
in  the  middle  ranges  with  the  curve  obtained 
from  the  100  cases  mentioned  above.  But 
for  the  lower  ranges  the  curve  gave  rather 
lower  mental  ages  than  the  new  line.  It  is  not 
improbable,  however,  that  the  curve  is  more 
trustworthy  here,  for  Terman  mentions  in  a 
footnote  that  the  mental  ages  given  in  his  table 
are  likely  to  be  somewhat  too  high. 

*  Cf.  p.  10  of  the  Manual  of  Directions  in  the  1921  American 
edition,  not  yet  obtainable  in  England. 

48 


DERIVATION  OF  MENTAL  AGES 
The  relation  between  the  original  line  ABK, 
the  curve  obtained  afterward  by  comparison 
with  the  S.B.  scale,  and,  finally,  the  line  plotted 


A/ 

^12.0 
0^ 

/ 

.d 

/y 

^/ 

2 

^    10. 

^ 

100 

160            £00 

Fig.  2 

from  the  data  given  by  Terman  is  represented 
in  Fig.  2,  which  is  drawn  only  approximately 
to  scale. 

In  Fig.  2  AA  represents  the  performance  of 

the  472  children  in  the  first  experiment  (ABK 

in  Fig.  i),  BB  the  curve  afterward  obtained  by 

4  49 


MENTAL  TESTS 

the  writer  from  the  comparison  of  the  scores 
of  the  100  children  with  their  S.B.  mental  ages, 
and  CC  the  line  plotted  from  Terman's  data 
referred  to  above. 
The  following  points  will  be  noted  : 
(i)  The  convergence  of  AA,  BB,  and  CC  as 
the  higher  scores  are  approached. 

(2)  The  coincidence  of  BB  and  CC  in  the 
middle  range. 

(3)  AA  and  BB  both  give  lower  mental  ages 
for  the  lower  scores  than  CC. 

Considering  AA,  BB,  and  CC  in  conjunction, 
and  allowing  so  far  as  possible  the  appropriate 
weight  to  each,  we  are  led  to  a  line  in  the  position 
of  the  dotted  line  in  the  figure  as  probably  a  close 
approximation  to  the  true  representation  of  the 
relation  between  mental  age  and  score  in  the 
Terman  group  test. 

When  this  line  was  plotted  and  its  equation 
expressed  in  the  form 

y  —  m%  +  c 
it  was  found  that,  compared  with  the  original  line 
AA,  m  had  diminished  from  {^  to  about  t%  or  f, 
while  c  had  increased  from  90  to  about  no. 
50 


DERIVATION  OF  MENTAL  AGES 

Thus  the  new  equation  will  be  very  nearly 

3;  =  1%  +  no 
so  that  mental  age  may  be  obtained  from  score 
by  the  formula 

Mental  age  in  months  =  (f  of  score  in 
Terman  group  test)  +110 

Notice  that  for  small  scores  the  first  item  on 
the  right  will  be  unimportant  compared  with 
the  second  term — the  constant  no ;  while  as 
the  score  increases  the  first  term  becomes  more 
and  more  important.  Thus  the  lower  mental 
ages  assigned  will  depend  very  largely  on  the 
value  given  to  c  (which  we  have  taken  as  no), 
while  the  higher  mental  ages  will  depend  more 
or  less  equally  on  c  and  m  (which  we  have  taken 
asf). 

Now  the  value  of  m  depends  on  the  slope  of 
the  line,  and  it  will  be  seen  from  Fig.  2  that  A  A 
and  CC  do  not  differ  greatly  in  slope,  while  the 
slope  of  BB,  though  it  varies  somewhat  from 
point  to  point,  is  on  the  whole  much  the  same 
as  that  of  A  A  and  CC.  For  A  A  m  =  tV,  for 
CC  m  =--  about  1^.    Therefore  there  is  unlikely  to 

51 


MENTAL  TESTS 

be  a  serious  error  in  the  value  we  have  taken  for 
m,  namely  |.  The  value  of  c,  on  the  other  hand, 
depends  on  the  value  of  y  for  which  x  vanishes  ; 
that  is,  it  represents  the  mental  age  in  months 
for  which  the  score  becomes  zero — the  age  below 
which  the  average  child  cannot  even  begin 
to  do  the  test.  For  AA  c  =  90,  a  mental  age 
of  7 J  years.  For  CC  c  =  about  116,  a  mental 
age  of  9  years  8  months.  There  is  a  considerable 
discrepancy  between  these  two  values  of  c,  and 
we  should  therefore  find  somewhat  serious 
differences  in  the  lower  mental  ages  according  as 
we  adopted  one  or  the  other. 

But  it  is  possible  to  get  nearer  to  the  true  value 
of  c.  For  it  was  pointed  out  that  the  formula 
y  ==  ^^  X  -\-  (^0  gives  mental  ages  which  are  too 
low,  and  we  saw  that  the  error  increases  as  the 
score  diminishes.  On  the  other  hand,  while  the 
mental  ages  found  from  CC  are  likely  to  be  too 
high,  the  comparison  here  is  at  any  rate  one 
between  score  and  true  or  S.B.  mental  age. 
The  value  116  for  c  is  thus  indicated  as  being 
considerably  nearer  the  truth  than  the  value  90. 
Moreover,  the  curve  BB  points  to  a  value  for 
52 


DERIVATION  OF  MENTAL  AGES 

c  of  about  io6,  or  a  mental  age  of  8  years  lo 
months.  It  is  therefore  improbable  that  there 
is  a  serious  error  in  the  value  we  have  provision- 
ally decided  to  take  for  c,  namely  no,  or  a 
mental  age  of  9  years  2  months.  But  it  will 
be  clear  that  the  precise  determination  of  this 
constant  is  important. 

While  the  formula  y  =  ^  x  +  no  probably 
gives  with  some  accuracy  the  correspondence 
in  general  between  true  mental  age  and  score  in 
the  Terman  group  test,  it  must  not  be  forgotten 
that  errors  may  occur  in  individual  cases.  If 
the  tests  were  perfect,  children  testing  at  the 
same  mental  age  on  the  S.B.  scale  should  score 
the  same  in  the  group  test.  But  in  practice 
it  not  infrequently  happens  that  children  of  the 
same  S.B.  mental  age  make  considerably  different 
scores  in  the  group  test.  The  reason  is  not  far 
to  seek.  In  the  group  of  100  children  who  were 
given  both  tests  all  cases  where  equal  or  nearly 
equal  mental  ages  were  correlated  with  different 
group  test  scores  were  carefully  analysed.  In 
nearly  all  such  cases  it  at  once  became  evident 
that  the  discrepancies  were  due  to  the  fact 

53 


MENTAL  TESTS 

that  the  two  tests  did  not  cover  exactly  the 
same  ground.  Certain  aspects  of  intelhgence 
to  which  some  of  the  S.B.  tests  are  directed 
are  not  touched  by  the  Terman  group  scale. 
For  example,  that  function  of  intelligence  which 
determines  control  of  visual  imagery  is  not 
specifically  tested  by  the  latter,  while  in  the  S.B. 
scale  there  are  at  least  two  tests  (No.  6  in 
year  xiv,  and  No.  2  in  '  superior  adult '  group) 
which  depend  primarily  on  the  manipulation 
of  visual  imagery.  Thus,  if  a  child  made  a 
lower  score  than  might  have  been  expected 
from  his  S.B.  mental  age,  it  was  found  on 
analysis  that  he  had  shown  up  well  when 
tested  by  the  S.B.  scale  in  some  direction, 
such  as  control  of  visual  imagery,  which  is  not 
specifically  tested  by  the  Terman  group  test. 

On  the  whole,  however,  the  distribution  of 
points  obtained  by  plotting  the  S.B.  mental 
ages  against  the  group  test  scores  shows  that 
the  S.B.  mental  ages  of  two  children  with  the 
same  score  are  much  more  likely  to  agree  fairly 
closely  than  to  differ  widely.  But,  at  the  same 
time,  it  is  clear  that  group  tests  should  be 
54 


DERIVATION  OF  MENTAL  AGES 

devised  in  such  a  way  as  to  make  them  as 
comprehensive  as  possible.  Otherwise  in  indi- 
vidual cases  different  scales  may  give  consider- 
ably different  results ;  and  we  cannot  feel 
complete  confidence  in  the  scientific  validity 
of  our  scales  as  trustworthy  instruments  of 
measurement  with  reference  to  an  objective 
standard  until  we  have  advanced  to  a  point 
where  different  scales,  when  applied  to  the  same 
child,  will  give  results  in  close  agreement.* 

1  The  writer  may  perhaps  be  permitted  to  refer  here  to  the 
Simplex  Group  Intelligence  Scale  (Harrap),  which  he  has  recently 
pubUshed,  and  which  is  devised  to  form  a  scale  sufficiently  com- 
prehensive as  regards  both  the  various  aspects  of  intelligence 
which  it  covers  and  the  age  range  of  the  children  to  whom  it  may 
be  suitably  applied.  The  Simplex  scale  was  administered  to  about 
a  hundred  children  whose  mental  ages  had  been  previously  estimated 
by  several  other  methods  (including  the  S.B.  individual  test).  The 
correlation  between  these  mental  ages  and  the  scores  in  the  Simplex 
test  was  found  to  be  ,94.  There  were  only  three  marked  dis- 
crepancies, a  proportion  so  small  that  it  might  well  have  been  due 
to  accidental  causes.  Neglecting  these  three  cases,  the  correlation 
rose  to  .97,  which  is  very  high  indeed. 


55 


CHAPTER  IV 

METHODS  OF  ESTIMATING  THE  TRUE  INTEL- 
LIGENCE QUOTIENTS  OF  ADULTS  AND 
ADOLESCENTS 

THE  results  of  experiment  seem  to  show 
that  inteUigence  grows  at  an  approxi- 
mately constant  rate  up  to  the  age  of 
something  over  fourteen  years.  We  may  term 
this  age  the  '  critical  age.'  Beyond  the  critical 
age  the  growth  of  intelligence  slows  down  until 
it  comes  to  a  complete  stop  at  the  age  of  some- 
thing over  seventeen  years.  The  age  at  which 
this  cessation  of  growth  occurs  will  be  called 
the  '  terminal  age.' 

As  we  have  seen,  a  convenient  way  of  express- 
ing the  absolute  level  of  intelligence  of  a  given 
child  at  a  given  time  is  by  means  of  '  mental 
age.'  The  rate  of  growth  of  intelligence  will 
then  be  the  ratio  of  the  mental  age  to  the  actual 
age.  This  ratio,  expressed  as  a  percentage,  is 
the  '  intelligence  quotient,'  and  is  found  to  be 

56 


INTELLIGENCE   QUOTIENTS 

approximately  constant  for  the  same  child  from 
early  childhood  up  to  the  critical  age,  though 
different  for  different  children.  The  I.Q. 
(which   really   represents   the   rate   of   mental 


Fig.  3 


growth)  is  thus  a  highly  convenient  index 
of  a  child's  degree  of  intelligence.  Fig.  3 
represents  the  growth  of  intelligence  of  three 
children,  one  inferior,  one  average,  and  one 
superior. 

It  is  by  no  means  certain,  though  for  conveni- 
ence it  has  been  assumed  in  Fig.  3,  that  the 

57 


MENTAL  TESTS 

critical  and  the  terminal  ages  are  approximately 
the  same  for  all  children.  ^ 

The  child's  degree  of  intelligence  is,  then, 
given  by  the  I.Q.,  which  represents  the  rate  of 
growth  during  the  period  when  that  rate  remains 
nearly  constant,  namely  below  the  critical  age. 
But  a  difficulty  arises  when  a  subject  is  not 
tested  until  the  adolescent  or  adult  period,  for, 
owing  to  the  slowing  down  of  mental  growth, 
the  ratio  of  mental  age  to  actual  age  will  change, 
and  will  therefore  no  longer  be  an  expression 
of  the  true  I.Q.,  but  will  enable  us  only  to  set 
certain  limits  to  the  subject's  intelligence. 

The  problem  before  us,  therefore,  is  as  follows  : 
How,  from  the  score  obtained  in  a  mental  test 
by  an  adult  or  an  adolescent  subject,  can  that 
subject's  true  I.Q.  be  estimated — i.e.,  the  I.Q. 
which  would  have  been  found  for  him  had  he 
been  tested  in  childhood  ? 

It  will  be  simpler  to  develop  the  methods  of 

^  Even  should  it  be  found  ultimately  that  in  the  case  of  all  children 
intelligence  ceases  to  grow  at  approximately  the  same  age,  different 
children  will  have  reached  at  that  age  very  different  levels  of  mental 
development.  E.g.,  the  dull  or  defective  child  may  have  attained 
a  mental  age  of  only  ten  or  twelve  years,  at  which  he  remains  for 
the  rest  of  his  life. 

58 


INTELLIGENCE  QUOTIENTS 

solution  proposed  by  reference  to  a  particular 
group  test  for  which  the  age  norms  of  perform- 
ance are  known.  For  this  purpose  we  shall  take 
the  Otis  group  test.      All  that  will  be  said  would 


Fig.  4 

be  equally  applicable,  nmtatis  mutandis,  to  any 
other  point-scale  test. 

I.  A  Direct  Method.  Fig.  4  represents  the 
curve  of  performance  in  the  Otis  group  test, 
as  plotted  from  the  table  of  norms.  ^  This  curve 
is  a  rough  reflection  of  the  curve  of  growth  of 
intelligence  of  the  '  average  '  child.  A  given 
child's  mental  age  is  found  by  taking  his  score  in 

^  See  Manual  of  Directions  for  Otis  group  test,  p.  65  (Harrap). 

59 


MENTAL  TESTS 

the  test  and  finding  from  the  curve  the  age  to 
which  it  corresponds.  The  percentage  ratio  of 
this  age  to  the  child's  actual  age  will  give  his  I.Q. 
A  glance  at  the  curve  in  Fig.  4  will  show  that  the 
critical  age  is  between  14  and  15  years  (actually 
14  years  5  months — see  table  of  norms)  and  the 
terminal  age  about  18  years.  Below  the  critical 
age  the  improvement  in  score  of  the  average  child 
is  I  point  per  month,  or  12  points  per  year. 

Let  us  first  consider  the  case  of  a  fairly  bright 
child  below  the  critical  age.^  Suppose  at  12 
years  of  age  he  scores  102.  His  mental  age  is 
then  13  years  2  months,  and  his  LO.  no.  From 
12  to  14  years  his  intelligence  will  continue  to 
grow  at  much  the  same  rate,  as  he  is  below  the 
critical  age.  At  14  years  his  mental  age  should 
therefore  be  15  years  5  months.  His  rate  of 
improvement  of  score  will  be  10  per  cent,  above 
that  of  the  average  child  (i.e.,  it  will  be  i.i  points 
per  month),  and  if  he  is  again  tested  at  14  years 
his   score  wiU    be    102  +  (24  x  i.i)— that  is, 

^  In  what  follows,  the  not  improbable  assumption  is  made  that 
the  intelligence  of  a  given  child  continues  to  bear  to  the  intelligence 
of  the  average  child  nearly  the  same  relation  after  the  critical  age 
as  before  that  age,  though  the  rate  of  growth  of  intelligence  of  both 
children  will  then  be  diminishing. 

60 


INTELLIGENCE   QUOTIENTS 

about  129.  From  the  curve  in  Fig.  4  the  age 
corresponding  to  a  score  of  129  is  about  17  years, 
and  the  ratio  of  this  to  14  years  will  thus  be 
greater  than  the  true  I.Q.  The  reason  for  this 
discrepancy  is,  of  course,  that  during  the  period 
12  to  14  years  the  child's  mental  growth  has  not 
been  following  a  curve  which  has  been  rising  less 
and  less  rapidly  like  the  one  in  Fig.  4  for  a  part 
of  the  time,  but  has  still  been  increasing  at  a 
comparatively  steady  rate.  Thus,  to  obtain  the 
true  I.Q.  we  must  extend  the  straight  portion 
of  the  curve  in  Fig.  4.  The  age  along  this 
straight  extension  corresponding  to  a  score  of 
129  will  be  15  years  5  months,  and  the  ratio  of 
this  to  14  years  gives  no,  the  true  I.Q.  In  other 
words,  when  the  child's  mental  age  rises  above 
the  critical  age,  we  must  compare  him  with  a 
fictitious  average  individual  ^  whose  intelligence 
(and  therefore  mental  age)  continues  to  grow  at 
a  steady  rate  indefinitely,  if  we  are  to  measure 
the  true  I.Q.  The  mental  age  thus  obtained  we 
shall  call  the  '  effective  mental  age.'  When  the 
actual  mental  age  is  below  the  critical  age  it 

1  Cf.  also  Manual  0/  Directions  for  Otis  group  test,  pp.  53  £f. 

61 


MENTAL  TESTS 

will  evidently  be  identical  with  the  effective 
mental  age.  Clearly  this  method  of  finding  the 
effective  mental  age  is  apphcable  to  a  subject 
of  any  age. 

The  above  case  points  the  way  to  the  solution 
of  the  main  problem  we  are  considering — the 
determination  of  the  true  I. Q.  of  the  adolescent 
or  adult.  In  such  cases  a  difficulty  arises  addi- 
tional to  that  found  in  the  case  of  the  bright  child 
just  below  the  critical  age.  For  what  are  we 
to  take  as  the  denominator  in  the  I. Q.  ratio  for 
subjects  over  the  critical  age  and  eventually 
over  the  terminal  age  ? 

We  may  consider  the  case  of  the  terminal  age 
first.  Let  us  now  suppose  that  intelligence, 
instead  of  slowing  down  after  the  critical  age  and 
gradually  coming  to  a  final  stop  at  the  terminal 
age,  continues  to  grow  at  the  same  constant  rate 
as  below  the  critical  age,  and  then  comes  to  a 
sudden  stop  at  the  same  level  as  is  reached  in 
actual  life  at  the  terminal  age.  Thus  we  must 
take  the  score  of  the  average  subject  in  the  test 
as  continuing  to  increase  by  i  point  per  month. 
Now  from  the  curve  in  Fig.  4  we  see  that,  for  the 
62 


INTELLIGENCE  QUOTIENTS 

average  individual,  the  '  terminal  score ' — i.e., 
the  score  reached  at  the  terminal  age — is  130. 
Had  the  score  still  increased  after  the  critical  age 
of  14  years  5  months  (at  which  the  score  is  117) 
at  the  rate  of  i  point  per  month,  the  score  of  130 
would  have  been  reached  at  the  age  of  15  years 
6  months.  We  may  call  this  age  the  *  effective 
terminal  age.'  Then  for  all  subjects  older  than 
the  terminal  age  we  must  take  the  effective  ter- 
minal age  as  the  denominator  of  the  I.Q.  ratio. 
The  I.Q.  for  such  subjects  will  therefore  be  the 
ratio  of  the  effective  mental  age  (calculated  as  pre- 
viously described)  to  the  effective  terminal  age. 
It  still  remains  to  deal  with  the  case  of  those 
subjects  who  are  younger  than  the  terminal  age, 
but  older  than  the  critical  age.  The  procedure 
here  is  to  find  the  age  at  which  the  score  corre- 
sponding on  the  curve  in  Fig.  4  to  the  actual 
age  of  the  subject  concerned  would  have  been 
reached  if  the  score  of  the  average  individual 
continued  to  increase  by  i  point  per  month  after 
the  critical  age.  The  age  thus  obtained  may  be 
termed  the  '  effective  age,'  and  must  be  used  as 
the  denominator  of  the  I.Q.  ratio  for  the  subject 

63 


MENTAL  TESTS 

concerned.  For  subjects  below  the  critical  age 
the  effective  age  will  evidently  be  identical  with 
the  actual  age,   while  for  subjects  above  the 


Fig.  5 

terminal  age  it  must  be  taken  as  equal  to  the 
effective  terminal  age. 

The  foregoing  may  perhaps  be  made  clearer 
by  an  example  illustrated  by  the  diagram  given 
in  Fig.  5. 

The  curve  ABC  in  Fig.  5  is  the  curve  of  per- 
formance in  the  Otis  test  as  in  Fig.  4.  BK 
64 


INTELLIGENCE   QUOTIENTS 

is  the  extension  of  the  straight  portion  AB  of 
the  curve.  B  is  at  the  critical  age,  C  at  the  ter- 
minal age,  and  M  at  the  effective  terminal  age. 
Suppose  a  child  of  just  15  years  makes  a  score 
of  145.  The  point  P  on  BK  gives  the  effective 
mental  age  as  16  years  9  months.  The  point 
Q  on  the  curve  ABC  corresponds  to  the  actual 
age  of  15  years,  at  which  the  average  score  is 
121.  QL  is  parallel  to  the  age  axis,  and  the 
point  L  on  BK  gives  the  effective  age  as  14 
years  9  months.  Thus  the  true  I.Q. — that  is, 
the  percentage  ratio  of  the  effective  mental  age 

^,       .  .16  years  9  months 

to  the  effective  age— is  ^4  years  9  months  ""  ^^^' 

i.e.,  ^^  X  100,  or  114. 

177 

The  procedure  for  finding  the  mental  age  of  an 
adult  or  adolescent  is  therefore  briefly  as  follows: 

From  the  subject's  score  obtain  the  effective 
mental  age  from  the  straight  line  ABK.  From 
the  curve  ABC  obtain  the  score  corresponding 
to  the  subject's  actual  age,  and  hence  from  the 
straight  line  ABK  the  effective  age  correspond- 
ing to  this  score.  ^     The  table  of  norms  can,  of 

^  Both  theory  and  practice  show  that  the  most  probable  form  of 

5  65 


MENTAL  TESTS 

course,  be  used  instead  of  the  curve,  provided 
we  remember  that  in  finding  the  effective  age 
and  the  effective  mental  age  the  average  score 
must  be  supposed  to  increase  at  the  same  rate 
above  the  critical  age  as  below  that  age.  The 
true  I.Q.  will  then  be  given  by  the  formula 

effective  mental  age  i  •  -i     •  t 

— -■ ^  X  100,  which  is  more  e^eneral 

etiective  age  '  ° 

than  the  older  formula  actual  aee  ^  ^^'^'  ^^^  ^^^^ 
latter  is  applicable  only  when  the  mental  age  and 
the  actual  age  are  both  below  the  critical  age. 

The  foregoing  cannot  be  applied,  except  in  a 
modified  form,  to  such  scales  as  the  Stanford 
Revision  of  the  Binet  tests.  In  the  latter  scale 
Terman  takes  the  effective  terminal  age  as  i6 
years,  and  arbitrarily  fixes  the  value  in  months 
of  the  tests  in  the  '  average  adult '  and  '  superior 

the  curve  of  performance  in  a  group  test  is  the  curve  kno\vn  as  an 
'  ogive.'  If,  however,  an  ogive  is  much  drawn  out  it  approximates 
closety  to  a  straight  line  over  a  considerable  range.  This  is  what 
usually  happens,  as  in  the  Otis  test.  If  the  data  permit  the  plotting 
of  the  ogive,  intercepts  for  finding  effective  mental  age  would  then 
be  made  on  the  latter  instead  of  on  the  simple  extension  of  the 
nearly  straight  portion  ;  but  the  difference  ^^411  be  practically 
important  only  for  scores  approaching  the  maximum  possible.  It 
should  be  noticed  that  the  drop  in  the  ordinary  curve  of  performance 
after  the  critical  age  is  due  to  the  slackening  growth  of  intelligence, 
and  not  to  the  tendency  toward  the  ogive  form. 

66 


INTELLIGENCE   QUOTIENTS 

adult '  groups  in  such  a  way  as  to  cause  adults  to 
test  on  the  average  at  a  mental  age  of  i6  years. 
The  method  outlined  here  could  thus  be  applied 
to  the  Stanford  Revision  only  after  an  investi- 
gation had  been  made  to  determine  the  actual 
average  performance  of  subjects  from  about  14 
to  (say)  20  years  of  age  in  the  upper  groups  of 
the  scale,  and  a  curve  of  performance  plotted 
from  the  results  of  this  investigation  apart  from 
any  arbitrarily  predetermined  value  in  months 
for  the  harder  tests. 

To  test  satisfactorily^  the  method  that  we  have 
been  discussing  it  would  be  necessary  to  retest 
the  same  children  at  intervals  during  adolescence 
in  order  to  discover  whether  the  I.Q.  calculated 
by  this  method  remained  nearly  constant  for 
the  same  child  and  equal  to  its  value  below  the 
critical  age.  Unfortunately,  such  retests  with 
the  Otis  scale  are  not  at  present  available,  but, 
in  their  absence,  artificial  cases  may  be  invented 
by  the  following  device  : 

A  child  of  given  age  and  given  score  at  that 
age  is  assumed.  Knowing  this,  the  distribution 
tables  at  the  end  of  the  Otis  Manual  of  Directions 

67 


MENTAL  TESTS 

will  enable  the  child's  percentile  rank  (P.R.)  to 
be  approximately  calculated.  On  the  basis  of 
this  P.R.,  examination  of  the  distribution  in  the 
other  age  groups  v/ill  make  it  possible  to  esti- 
mate this  same  child's  probable  score  at  other 
ages,  since  his  P.R.  remains  constant,  due 
allowance  being  made  for  the  fact  that  the  higher 
age  groups  are  not  fullj^  representative.  Thus, 
for  example,  if  his  P.R.  is  70,  to  find  his  probable 
score  at  12  years  of  age  look  at  the  distribution 
of  scores  for  the  age  group  12,  and  find  the  score 
of  P.R.  70. 

Two  examples  may  be  given  in  illustration, 
one  of  a  bright  and  one  of  a  dull  child.  ^ 

(i)  A  bright  child  who  at  16  makes  a  score  of 
180.  The  probable  scores  of  this  child  for 
various  ages  were  found,  and  hence  his  effective 
mental  ages,  which,  with  the  effective  ages,  gave 
the  values  of  his  LQ. 

The  results  are  given  in  the  table  on  p.  69. 

Hence  for  this  case  the  method  gives  a  nearly 
constant    LQ.     Note    that    by    the    ordinary 


1  These  two  examples  were  suggested  to  the  writer  (by  Mr  Burt) 
as  a  test  of  his  method. 

68 


INTELLIGENCE   QUOTIENTS 

method  this  child  would  at  12  be  supposed  to 
have  a  mental  age  of  18  (130  being  the  average 
score  at  18),  giving  an  I.Q.  of  150,  very  wide 
of  the  mark.     Above   12   years   it   would   not 


Actual 
Age  of 
Child 
(Years) 

Probable 
Score 

Effective 

Mental 

Age 

(Months) 

Average 

Score  at 

Child's 

Actual  Age 

Effective 

Age 
(Months) 

True  I.Q. 

18 

182 

238 

130 

186 

f|fXIOO  =  I28 

16 

180 

236 

125 

181 

Iff  X  100=130 

15 

173 

229 

122 

178 

f^-|XIOO  =  I29 

14 

160 

216 

112 

168 

f^|x  100=129 

12 

130 

iSo 

88 

144 

fHx  100=132 

10 

104 

160 

64 

120 

i|gx  100=133 

be  possible  to  assign  him  a  mental  age  by  the 
ordinary  method. 

(2)  A  dull  child  of  I.Q.  about  70.  This  child's 
mental  age  when  12  years  old  would  then  be 
about  8  years  5  months,  and  therefore  his  score 
would  be  45  (see  table  of  norms). 

The  results  are  given  in  the  table  on  p.  70. 

Here  the  variation  of  I.Q.  is  too  great  to  be 
satisfactory,  but  it  should  be  noted  that  the 

69 


MENTAL  TESTS 

probable  scores  obtained  for  this  child  by  the 
P.R.  method  for  ages  above  14  are  certainly 
considerably  higher  than  his  scores  would  actu- 
ally be  in  practice.     For  the  probable  scores 


Actual 
Age  of 
Child 
(Years) 

Probable 
Score 

Effective 

Mental 

A-e 

(Months) 

Avernce 

Score  at 

Child's 

Actual  Age 

Effective 

Age 
(Months) 

True  I  Q. 

10 

30 

86 

64 

120 

xVoXIOO=    72 

12 

45 

lOI 

88 

144 

l£iXIOO=    70 

14 

68 

124 

112 

168 

Hi  X 100=  74 

15 

81 

137 

122 

178 

iff  X  100=    77 

16 

90 

146 

125 

181 

i||XI00=   80 

18 

99 

155 

130 

186 

i||xioo=  83 

make  his  rate  of  increment  of  score  after  14 
greater  than  that  of  the  average  child,  which 
would  clearly  not  be  the  case.  Hence  in  practice 
our  method  would  give  an  I.Q.  constant  within 
limits  considerably  narrov/er  than  is  apparent 
from  the  example.  It  simply  happens  that  the 
P.R.  method,  which  is  here  an  artificial  device 
dependent  for  its  accuracy  on  the  accuracy 
and  comprehensiveness  of  the  distribution 
70 


INTELLIGENCE  QUOTIENTS 

tables,   does  not,   in  this  case,   give  probable 
scores  sufficiently  near  to  the  true  scores. 

Now  let  p  stand  for  the  child's  score  and  q 
for  the  average  score  at  his  age.  Then  it  will  be 
seen  from  the  above  examples  that  his  effective 
age  is  ^  +  56,  and  his  effective  mental  age 
p  +  56,  so  that  his  true  I.Q.  is  ^-±^  x  100. 
Similarly,  for  the  Terman  group  test  we  should 
have,  in  consequence  of  the  formula  obtained 
in  the  last  chapter,  true  I.Q.  =  f^  +  "^  x  100. 
Evidently  it  would  be  quicker  and  simpler  to 
find  the  true  I.Q.s  from  formulae  like  these  than 
direct  from  the  curve  as  in  Fig.  5.  But  the  in- 
troduction and  discussion  of  the  latter  was 
necessary  to  make  the  principle  of  the  method 
clear.  As  a  matter  of  fact,  in  practice,  once  the 
norms  have  been  established  (and  hence  the 
critical  and  terminal  ages)  for  any  group  test, 
the  best  thing  to  do  would  be  to  construct,  by 
means  of  a  formula  like  the  above,  a  ready- 
reckoner  giving  true  I.Q.  direct  from  score  and 
actual  age. 

The  results  we  have  arrived  at  may  perhaps 

71 


MENTAL  TESTS 

be  best  summarized  by  a  recapitulation  of  the 
definitions  of  the  new  terms  used. 

The  critical  age  is  the  age  at  which  intelHgence 
ceases  to  grow  at  an  approximately  constant 
rate,  and  its  development  begins  to  slow  down. 

The  terminal  age  is  the  age  at  which  intelli- 
gence stops  growing  altogether. 

The  terminal  score  is  the  average  score  reached 
at  the  terminal  age. 

The  effective  terminal  age  is  the  age  at  which 
the  terminal  score  would  be  reached  if  intelli- 
gence continued  to  grow  after  the  critical  age 
at  the  same  rate  as  before  the  critical  age. 

The  effective  mental  age  of  a  subject  is  the  age 
at  which  the  score  he  makes  would  be  reached 
by  him  if  his  intelhgence  continued  to  grow 
indefinite^  at  the  same  rate  as  below  the 
critical  age.  When  the  effective  mental  age  is 
less  than  the  critical  age  it  is  identical  with  the 
actual  mental  age. 

The  effective  age  of  a  subject  is  the  age  at  which 
the  average  score  corresponding  to  his  actual 
age  would  be  reached  if  intelligence  continued 
to  grow  at  the  same  rate  after  the  critical  age  as 
72 


INTELLIGENCE  QUOTIENTS 

before  that  age.  If  the  effective  age  is  less  than 
the  critical  age  it  will  be  identical  with  the  actual 
age  ;  if  it  is  greater  than  the  terminal  age  it  must 
be  taken  as  equal  to  the  effective  terminal  age. 

The  general  formula  for  the  I.Q.  then  be- 
comes the  percentage  ratio  of  the  effective 
mental  age  to  the  effective  age.  This  formula 
gives  the  true  I.Q.  for  adults  and  adolescents — 
that  is,  the  I.Q.  which  characterized  them  when 
they  were  below  the  critical  age. 

For  the  Otis  group  test  the  critical  age  is 
14  years  5  months,  the  terminal  age  18  years, 
and  the  effective  terminal  age  15  years  6  months. 
It  will  be  important  to  determine  whether  the 
critical,  terminal,  and  effective  terminal  ages 
are  (a)  approximately  the  same  for  different 
children,  (h)  approximately  the  same  for  dif- 
ferent test  scales. 

II.  An  Indirect  Method.  The  writer  has 
also  obtained  estimates  of  true  I.Q.s  by  means 
of  a  percentile  rank  method.  This  method 
rests  on  two  facts  : 

(i)  The  percentage  distribution  of  I.Q.s  in  any 
particular  age  group,  if  sufficiently  large,  will 

7Z 


MENTAL  TESTS 

approximate  closely  to  the  percentage  dis- 
tribution of  I.Q.s  for  the  whole  population. 
This  simply  means  that  any  two  large  age 
groups  will  probably  have  about  the  same 
distribution  of  LQ.s. 

(2)  Within  a  given  age  group  the  distribution 
of  I.Q.s  will  be  the  same  as  that  of  the  mental 
ages,  since  the  denominator  of  the  LQ.  ratio 
will  be  approximately  the  same  for  every 
individual  in  the  group.  The  narrower  the 
limits  of  age  between  which  the  group  is  taken, 
the  nearer  will  the  agreement  be. 

In  order  to  use  this  method  two  things  must 
be  known  : 

{a)  The  general  distribution  of  I.Q.s  below 
the  critical  age  (these  being  true  I.Q.s). 

[b)  The  distribution  of  scores  (and  therefore 
of  mental  ages)  in  an  age  group  with  a  narrow 
age  range  about  the  age  of  the  particular  indivi- 
dual who  is  being  dealt  with. 

The  first  is  now  fairly  well  known,  as  is  also  the 
second  in  the  case  of  some  group  tests. 

The  method,  which  is  simple  enough  to  apply, 
needs  only  a  brief  description.  Knowing  the 
74 


INTELLIGENCE  QUOTIENTS 

actual  age  of  the  subject  concerned,  and  the 
score  he  has  made,  find  from  the  distribution 
tables  the  P.R.  of  his  score,  and  therefore  of  his 
mental  age,  in  an  age  group  of  individuals  of 
nearly  the  same  age  as  himself.  As  pointed  out 
above  in  (2),  the  P.R.  of  his  mental  age  within 
this  age  group  will  be  the  same  as  the  P.R.  of 
his  true  I.Q.  in  the  age  group,  and  the  last 
is  the  same  as  the  P.R.  of  his  true  I.Q.  when  he 
was  in  any  other  age  group  (see  (i)  above).  Now 
the  distribution  of  true  I.Q.s  is  known  from  the 
age  groups  below  the  critical  age.  Hence, 
knowing  the  P.R.  of  the  given  subject,  his  true 
I.Q.  can  be  found  at  once. 

For  example,  suppose  a  person  of  something 
over  16  years  of  age  makes  a  score  of  162  in 
the  Otis  test.  His  P.R.  in  the  16-17  age  group, 
found  from  the  distribution  table  at  the  end  of 
the  Otis  Manual,  is  about  75.  Now  in  the 
general  distribution  of  true  I.Q.s  the  75  per- 
centile is  about  109.  Hence  this  person's  true 
I.Q.  is  about  109.  For  greater  accuracy,  how- 
ever, the  age  ranges  should  be  narrower  than 
those  in  the  Otis  distribution  tables. 

75 


MENTAL  TESTS 

The  method  we  have  been  discussing  is 
applicable,  with  slight  modifications,  to  the 
Stanford-Binet  individual  scale,  as  explained 
in  the  appended  note. 


NOTE  ON  A  METHOD  OF  ESTIMATING  THE 
TRUE   STANFORD-BINET  I.Q.s   OF  ADULTS 

The  following  method  of  estimating  the  true 
Binet  (Stanford  Revision)  I.Q.s  of  adults,  or  of 
children  approaching  i6  years  of  age,  which  has 
occurred  to  the  writer  in  the  course  of  his 
general  inquiry  into  methods  of  estimating  the 
true  LQ.s  of  adults  and  adolescents,  may  perhaps 
be  of  interest  to  those  engaged  on  Binet  tests. 

The  difficulty  which  occurs  in  this  connexion 
with  the  Binet  scale  lies  in  the  fact  that,  owing 
to  lack  of  extensiveness  in  the  upper  ranges, 
LQ.s  of  adults,  as  measured  by  the  scale,  are 
bound  to  drop  below  their  true  values.  The 
greatest  possible  I.Q.  at  which  an  adult  can 
test  is  about  126. 

For  the  method  to  be  used  two  things  must 
be  known  :  (i)  the  distribution  of  children's 
76 


INTELLIGENCE   QUOTIENTS 

Binet  I.Q.s  ;  (2)  the  distribution  of  the  mental 
ages  of  adults  as  measured  in  the  ordinary 
way  by  the  Binet  scale. 

The  first  is  fairly  well  known  (cf.  Terman, 
The  Measurement  of  Intelligence  (Harrap),  pp. 
66  and  78),  but  there  are  not  yet  sufficient  data 
to  obtain  the  second  accurately,  though  Terman 
gives  a  diagram  based  on  62  '  normal '  cases 
on  p.  55  of  the  book  referred  to.  The  writer 
has,  however,  found  the  latter  useful  as  a  rough 
working  basis.  (N.B.  In  Terman's  diagram  on 
p.  55,  15. II  seems  to  be  a  misprint  for  16. 11.) 

To  find  true  Binet  I.O.s  proceed  thus  :  Hav- 
ing obtained  the  mental  age  of  any  particular 
adult  in  the  ordinary  way,  find  his  P.R.  from 
the  distribution  table  for  mental  ages  of  adults. 
vSince,  in  finding  the  I.O.,  the  actual  age  is 
taken  for  every  adult  as  16  years,  the  P.R. 
obtained  will  also  be  the  given  person's  P.R. 
as  regards  I.Q.  Then  from  the  distribution 
tables  for  children's  I.Q.s,  which  are  true  I.Q.s 
(the  distribution  remaining  the  same  for 
different  age  groups),  it  is  possible  to  find 
directly   the   approximate  value  of   the  given 

77 


MENTAL  TESTS 

adult's  true  I.Q.,  his  P.R.  having  been  found 
as  explained  above. 

In  conclusion,  two  points  should  be  noted  : 
(i)  for  I.Q.s  below  about  105  the  value  ob- 
tained in  the  ordinary  way  is  probably  very 
nearly  the  true  one  ;  and  (2)  Terman's  distri- 
bution diagram  for  adults  can  be  used  only  up 
to  mental  ages  of  about  18  years  6  months. 
When  more  extensive  data  are  available  for 
obtaining  this  distribution  the  percentile  rank 
method  should  give  I.Q.  values  of  consider- 
able accuracy. 


Table  of  Provisional  Values  obtained 
BY  P.R.  Method 


Mental  Age  found 
by  Usual   Method 

True  I 

16  years  6  months 

107 

16 

,     9       .. 

108 

17 

,     0 

IIO 

17 

,     3       .. 

112 

17 

.     6       „ 

115 

17 

,     9       •' 

118 

18 

,     0 

122 

18 

,     3       .. 

126 

18 

,     6       „ 

132 

78 


CHAPTER  V 

THE  RELIABILITY  OF  THE  GROUP   INTELLI- 
GENCE TEST  AS  AN  INDEX  OF  EDUCABILITY 

IN  Chapter  II  we  discussed  an  experiment 
the  results  of  which  pointed  to  the  fact 
that  the  Stanford-Binet  scale  provided  an 
accurate  means  of  gauging  the  educable  capacity 
of  a  child.  This  scale  is,  however,  administered 
individually,  and  thus  involves  the  expenditure 
of  a  considerable  amount  of  time,  especially 
when  large  groups  of  children  are  to  be  tested. 
In  practice,  therefore,  it  is  necessary  to  employ 
group  tests  which  can  be  apphed  to  a  great 
many  children  simultaneously,  retaining  the  in- 
dividual scale  as  a  standard  of  reference  and  a 
source  of  further  information  in  particular  cases. 
The  question  therefore  arises  as  to  the 
reliability  of  the  group  test  as  an  index  of 
educability.  There  are  two  methods  of  ap- 
proaching the  problem,  one  indirect,  the  other 
direct.  The  first  method  consists  in  comparing 
the  results  of  the  group  test  with  those  of  the 

79 


MENTAL  TESTS 

individual  test,  the  reliability  of  which  has 
already  been  largely  established.  The  second 
method  consists  in  comparing  the  results  of 
the  group  test  with  those  of  ordinary  educa- 
tional tests  which  may  be  taken  as  affording 
reasonably  accurate  estimates  of  the  educable 
capacities  of  children  who  have  been  at  school 
for  some  years.  Experiments  illustrating  both 
methods  will  now  be  described. 

L  The  Indirect  Method.  The  group  of 
children  tested  individually  by  the  S.B.  scale, 
as  described  in  Chapter  II,  was  afterward 
given  a  typical  group  intelligence  test.  It 
should  be  noted  that  the  group  tests  at  present 
in  vogue,  though  they  differ  from  one  another 
in  detail  as  regards  both  particular  forms  of 
tests  employed  and  particular  items  of  different 
tests,  conform  largely  to  a  type.  If  the 
reliability  of  such  a  typical  group  test  can  be 
established,  it  therefore  follows  that  any  group 
test  of  similar  type,  provided  it  is  reasonably 
comprehensive,  and  provided  it  has  been  shown 
to  be  sufficiently  suitable  by  trial,  is  Hkely  to  be 
equally  reliable, 
80 


THE  GROUP  INTELLIGENCE  TEST 

In  the  case  with  which  we  are  deahng  the 
figures  obtained  from  the  comparison  of  the 
results  of  the  individual  and  the  group  tests 
were  considered  for  each  batch  of  children 
separately  (see  Chapter  II)  as  well  as  for  the 
group  as  a  whole. 

The  rank  correlations  for  the  five  batches  be- 
tween scores  in  the  group  test  and  mental  ages 
obtained  by  the  individual  test  were  as  follows  : 

Group  A  Group  B  Group  C  Group  D  Group  E 
.86     .86     .91     .88     .92 

When  the  group  was  taken  as  a  whole  the 
rank  correlation  worked  out  at  .89. 

It  will  be  seen  that  these  correlations  are 
high,  and  it  may  therefore  be  safely  concluded 
that  the  group  test  is,  on  the  whole,  a  sufficiently 
reliable  index  of  educability. 

On  examining  the  results  in  detail  two  con- 
clusions were  drawn  : 

(i)  The  group  test  is  a  very  good  method  of 
ordering  a  number  of  children  in  respect 
of  intelligence,  with  sufficient  accuracy. 

(2)  The  group  test  is  not  so  reliable  as  the 
6  81 


MENTAL  TESTS 

individual  test  for  assigning  the  exact 
mental  level  of  a  particular  child.  Though 
in  general  the  agreement  between  the 
two  tests  is  close,  in  particular  cases 
the  group  test  may  rate  a  child  too  low 
or  too  high. 

The  factor  which  leads  to  the  second  con- 
clusion seems  to  be  that  the  group  tests  at 
present  in  use  are  not  so  comprehensive  as  the 
individual  test  in  the  appeal  they  make  to 
the  intelligence.  The  mental  capacity  of  each 
child  tested  in  the  experiment  here  described 
was  analysed  as  far  as  possible  under  thirteen 
heads/  all  of  which  appear  to  be  tested  speci- 
fically by  the  S.B.  scale,  but  some  of  which  are 
not  tested  specifically  by  the  group  scale.  In 
the  cases  of  discrepancy  scrutiny  of  the  analysis 
revealed,  almost  invariably,  that  the  child  had 

^  These  were  :  (i)  Readiness  and  ability  in  applying  knowledge  ; 
(2)  Power  to  discriminate  essentials  ;  (3)  Richness  and  logical 
integrity  of  the  associative  processes  ;  (4)  Ability  to  control  and 
concentrate  attention  ;  (5)  Power  of  comprehension  ;  (5)  Ability 
to  hold  in  mind  the  conditions  of  a  problem  ;  (7)  Ingenuity  and 
practical  judgment  ;  (8)  Steadiness  of  purpose  ;  (9)  Power  of 
forming  abstract  ideas  ;  (lo)  Power  of  generalization  ;  (11)  Critical 
ability  ;  (12)  Ability  to  manipulate  imagery  (especially  visual 
imagery)  ;    (13)  Development  of  social  consciousness. 

82 


THE  GROUP   INTELLIGENCE  TEST 

shown  up  well  (or  badly,  as  the  case  might  be) 
in  the  individual  test  in  certain  directions  which 
were  not  specifically  covered  by  the  group  test. 
The  remedy  lies,  of  course,  in  making  our  group 
scales  so  comprehensive  that  we  may  be  assured 
of  their  equal  reliability  with  the  individual  scale, 
not  only  in  general,  but  also  in  particular  cases.* 

11.  The  Direct  Method.  In  Chapter  III 
an  experiment  was  described  in  which  some 
500  children  were  tested  by  the  Terman  group 
intelligence  scale,  and  the  method  of  deriving 
the  children's  mental  ages  and  I.O.s  from  their 
scores  in  the  test  was  explained. 

Now  these  children  had  also  been  tested  by 
an  ordinary  written  examination  in  arithmetic 
and  English,  so  that  data  were  available  for 
comparing  the  results  of  the  intelligence  test 
with  those  of  the  ordinary  educational  test. 

Before  the  comparison  could  be  effected  it  was 
necessary  to  express  the  results  of  the  written 
examination  in  terms  similar  to  those  in  which 
the  results  of  the  intelligence  test  were  expressed. 
This  was  done  by  means  of  a  procedure  exactly 

^  C/.  footnote  at  eud  oi  Chapter  III. 

83 


MENTAL  TESTS 

analogous   to   that   described  in   Chapter   III. 
That  is,  the  age  norms  of  performance  in  the 
written    examination    (taking    the    arithmetic 
and    Enghsh    together)    were    estabhshed    by 
finding  the  average  scores  of  the  children  in  the 
age  groups  loj,  lof,  iij,  iij,  i2i,  i2|  years. 
A  curve  of  performance  was  then  obtained,  and 
by  comparing  the  score  of  a  particular  child 
with  this  curv^e  it  was  possible  to  assign  to  him 
a  '  mental  age  '  indicative  not  simply  of  native 
intelligence,    but    of    educational    attainment. 
Dividing  this  quantity  by  the  child's  actual  age, 
and  expressing  the  result  as  a  percentage,  a  figure 
was  obtained  which  might  be  fairly  taken  as  an 
index  of  the  child's  educability,  and  was  there- 
fore termed  the  '  educability  quotient '  (E.Q.).^ 
At  this  point  it  should  be  noted  that  the 
papers  in   the  written  examination  were  cor- 
rected in  the  ordinary  way  by  a  single  examiner. 
A   subjective   factor   was   thus   involved   of   a 
kind  which  did  not  enter  into  the  intelligence 
test,  the  scoring  of  which  was  purely  mechanical 
and  independent  of  any  particular  examiner. 

'  C/.  Chapter  II.  p.  29. 

84 


THE  GROUP  INTELLIGENCE  TEST 

It  was  therefore  to  be  expected  that  the  results 
of  the  written  examination  would  not  exhibit 
the  same  regularity  as  those  of  the  intelligence 
test.  This  was  in  fact  the  case  ;  nevertheless 
the  results  were  sufficiently  regular  to  render  the 
fitting  of  a  curve  of  performance  to  the  data 
a  matter  of  little  doubt  or  difficulty. 


■ 

Percentage  of  Children 

Boys                  Girls 

E.g.  below  I.Q. 

E.O.  and  I.Q.  corresponding 

E.Q.  above  I.Q. 

24.2                  16.3 

557            56.1 
20.1            27.6 

1 

Now  it  is  reasonable  to  suppose  that  if  a  child 
is  educated  in  sufficiently  favourable  circum- 
stances his  attainment  should  keep  pace  with 
his  intelligence,  so  that,  provided  our  intelli- 
gence tests  are  reliable,  a  close  correspondence 
should  appear  between  the  I.Q.s  and  the  E.O.s. 

The  results  of  this  experiment  are  given  in  the 
above  table  (I.Q.  and  E.O.  are  said  to  correspond 
when  they  are  within  five  points  of  one  another). 

These  figures  speak  for  themselves,  and  it  is 

85 


MENTAL  TESTS 

evident  that  the  agreement  between  the  results 
of  the  two  kinds  of  test  is  strikingly  close  in  a 
very  considerable  proportion  of  cases.  Com- 
plete agreement  could  not  in  any  case  be 
expected,  for  many  causes  may  mihtate  against 
close  agreement  between  the  I.Q.  and  E.Q.  of 
a  particular  child.  Past  or  present  illness, 
absence  from  school,  late  entry,  bad  teaching, 
laziness,  will  all  tend  to  cause  the  E.Q.  to  drop 
below  the  I.Q.  ;  whereas  a  temperament  more 
industrious  than  the  average,  good  teaching, 
and  similar  causes  may  make  the  E.Q.  rise  above 
the  I.Q.,  it  being  remembered  that  these  two 
indexes  are  obtained  by  comparison  with  the 
average.  In  this  experiment,  of  course,  the 
average  was  not  that  of  a  sample  of  the  whole 
population,  but  of  this  particular  group  of 
children,  which  (as  explained  in  Chapter  III) 
was  somewhat  above  the  average  of  the  whole 
population.  In  obtaining  the  I.Q.  it  was  of 
course  necessary  to  use  the  first  formula  given  in 
Chapter  III. 

The  distributions  were  given  by  the  following 
figures  : 
86 


THE  GROUP  INTELLIGENCE  TEST 


E.Q. 

No.  of 
Boys 

No.  of 

Girls 

Total 

Under  80     .         .         . 

9 

25 

34 

81-90 

28 

70 

98 

91-100 

49 

87 

136 

lOI-IIO 

55 

67 

122 

111-120 

19 

30 

49 

I 21-130 

9 

12 

21 

131  and  over 

6 

6 

12 

175 

297 

472 

i.Q. 

No.  of 
Boys 

No.  of 
Girls 

Total 

Under  80     . 

19 

35 

54 

81-90 
91-100 

27 
41 

73 
85 

100 
126 

lOI-IIO 

111-120 

40 
24 

57 
28 

97 
52 

121-130 
131  and  over 

15 

8 

17 
2 

32 
10 

174 

297 

471 

These  figures  give  a  median  of  just  under 
100  for  both  I.Q.  and  E.O.,  but  the  semi-inter- 
quartile range  shows  a  shghtly  greater  scatter 
for  the  former  than   for  the  latter,   a   result 

87 


MENTAL  TESTS 

similar  to  that  found  in  the  experiment  de- 
scribed in  Chapter  II.  It  probably  points  to  a 
certain  insufficiency  in  the  elasticity  of  pro- 
motion so  far  as  the  brighter  children  are 
concerned. 

Summing  up,  it  may  be  said  that  the  results 
thus  obtained,  both  by  the  indirect  and  by  the 
direct  methods,  afford  good  grounds  for  regard- 
ing tests  of  the  group  scale  type  as  useful  and 
sufficiently  accurate  means  of  estimating  educa- 
bility,  while  this  reliability  is  likely  to  be 
increased  when  our  tests  are  so  devised  as  to 
probe  the  child's  intelligence  from  as  many 
directions  as  possible. 


88 


CHAPTER  VI 

CONCLUSION 

WE  may  now  consider,  very  briefly,  the 
conclusions  to  which  we  are  led  by 
the  results  of  the  experiments  which 
have  been  described.  In  the  first  place,  it  will 
probably  be  admitted  that  these  experiments  go 
some  way  toward  establishing  the  reliability 
of  intelligence  tests  in  carrying  out  the  special 
purpose  for  which  they  were  designed — the 
separation  and  grading  of  children  according 
to  degrees  of  educability.  Indeed,  the  general 
correspondence  between  tlie  children's  per- 
formances in  the  tests  and  the  levels  of  educa- 
tional attainment  which  they  had  shown  them- 
selves capable  of  reaching  was  really  very 
striking  indeed.  It  is  therefore  evident  that 
intelligence  tests  should  no  longer  be  regarded 
with  suspicion  as  a  mere  *  stunt,'  but  should 
be  recognized  as  a  valuable  instrument  capable 
of  affording  great  aid  in  the  advancement  of  the 

89 


MENTAL  TESTS 

theory  and  the  practice  of  educational  science. 
The  task  of  the  future  is  to  perfect  them. 

Secondly,  it  must  not  be  forgotten  that 
intelligence  is  not  the  only  factor  necessary  for 
success  in  school  and  in  after  life,  though  it  is 
the  most  important.  Unless  he  possesses  a 
certain  minimum  of  intelligence  the  child  will 
not  be  capable  of  profiting  by  advanced  educa- 
tion ;  but  even  if  he  exceeds  this  minimum 
other  factors  are  necessary  to  his  successful 
progress.  Not  only  must  he  reach  a  certain 
standard  of  physical  fitness,  but  his  tempera- 
ment and  character  must  be  such  that  he  is 
impelled  to  use  his  intelligence  effectively.  It 
is  just  here  that  the  real  value  of  the  ordinary 
educational  tests  of  attainment  becomes  ap- 
parent. For  we  want  to  know  not  only  whether 
a  child  is  intelligent,  but  also  whether  he  is 
capable  of  making  good  use  of  his  intelligence. 
Intelligence  tests  are  therefore  most  valuable 
when  their  results  are  scrutinized  in  comparison 
with  the  results  of  educational  tests  together 
with  records  of  school  careers.  But  if  our 
educational  tests  are  to  attain  their  full  value 
90 


CONCLUSION 
an  attempt  must  be  made  to  standardize  them 
as  effectively  as  intelligence  tests  are  stan- 
dardized, and  also  to  interpret  their  results,  in 
the  case  of  each  child,  by  an  appropriate  age 
allowance  carried  out  by  some  such  method 
as  that  described  in  Chapter  V.  Unless  dealt 
with  in  this  way  educational  tests  lose  much  of 
their  significance.  All  cases  in  which  a  marked 
discrepancy  is  found  between  the  results  of  the 
intelligence  and  of  the  attainment  tests  respec- 
tively should  be  made  the  subject  of  further 
study. 

Finally,  a  word  may  be  said  as  to  the  adminis^ 
tration  of  the  tests.  While  group  testing  can  be 
carried  out  by  any  reasonably  competent  person 
who  is  wilUng  to  take  the  trouble  to  follow 
exactly  the  prescribed  procedure  (which  is 
becoming  more  and  more  simple  as  the  tests  are 
improved),  it  is  probable  that  the  administra- 
tion of  the  individual  tests  should  be  left  to 
persons  having  some  expert  knowledge  of  the 
subject.  In  any  case  detailed  interpretation  of 
results  is  a  matter  for  the  psychologist.  For 
this  reason,  as  well  as  on  account  of  the  time 

91 


MENTAL  TESTS 

factor,  practical  testing  in  the  future  will  in- 
evitably (and  quite  rightly)  be  carried  out 
mainly  by  the  group  method.  The  individual 
scale  will  remain  as  an  ultimate  standard  of 
reference,  to  which  recourse  will  be  had  for 
purposes  of  comparison,  and  for  dealing  with 
especially  interesting  and  difficult  cases,  whether 
supernormal,  subnormal,  or  abnormal. 


92 


INDEX 


Administration  of  tests,  91 

Age,  mental,  17  f.,  27,  40  ff., 
56  £E.;  critical,  56,  58,  63,  72; 
terminal,  56,  58,  72  ;  effec- 
tive, 63  f.,  72  ;  effective 
mental,  61  f.,  72  ;  effective 
terminal,   63,   72 

Age  performance,  in  arith- 
metic, 27,  83  f.  ;  in  mental 
tests,  41,  43  ff.,  71  ;  in 
composition,  28  ff.,  71 

American  tests,  26,  36 

Aptitude,    special,    14 

Arithmetic,  25,  3(3  f.  ;  quo- 
tient, 27  ff. 

Attainment,  16,  34  f.,  90 

Attention,  13 

Ballard,  P.  B.,  27  f. 

Binet,  Alfred,  24 

Bravais,  33 

British  Journal  of  Psychology, 

5 
Brown,  W.,  21  «. 
Burt,  Cyril,  68 

Character,  90 
Coaching,    for    mental    tests, 
II  f. 


Composition,  25,  27  f.,  36  ff.  ; 

quotient,  29  ff. 
Correlation,  20  f.,  29  ff,,  81 
Critical  age,  56,  63,  72 
Criticisms    of    mental    tests, 

9fif. 
Curve  of  performance,  43  ff., 

59  ff.,  64  ff. 

Defectiveness,  mental,  32  f. 

Discrepancy  between  results 
of  different  tests,  31,  53  f. 

Distribution,  34  ;  of  intelli- 
gence quotients,  74  ff.,  87  ; 
of  educability  quotients, 
87  ;   normal,  43  w. 

Educability,    13,    16,   24  ff., 

29  ff.,  79  ff.,  81,  84  ff.,  89 
Effective  age,  63  f.,  72 
Effective  mental  age,  61  f.,  72 
Environment,   16 
Error,  probable,  20 
Examinations,   83  ff.,   go 
Experience,   15 

Factor,  general  intelligence, 

13  f.  ;  subjective,  17,  84  f. 

Formula  for  mental  age,  46  ff. 

93 


MENTAL   TESTS 


General  intelligence  factor, 

13  f. 
Growth  of  intelligence,   14  f., 

57fl[. 

Heterogeneity  of  material, 
30  n. 

Imagery,  visual,  54 
Intelligence,      definition      of, 

I2f.  ;  growth  of,  14  f.,  57  £f.  ; 

quotient,  17  f.,  26  ff.,  56  ff.; 

aspects  of,  42,  82 
Interest,    14 
Interquartile  range,  19 

Knowledge,  15 

Mean  square  deviation,  18  f. 

Median,  19 

Mental  defectiveness,   32  f. 

Normal  distribution,  43  n. 
Norms,  12,  44,  84 

Objective  standard,  17,  22 
Ogive,  44  n.,  65  n. 
Otis  group  test,  59  fi. 

Pearson,  K.,  29,  33 
Percentile  rank,  18,  68,  74  ff. 
Performance,  curve  of,  43  ft"., 
59  ff.,  64  ff. 


Personal  equation,  17,  84  f. 
Presupposition      of      mental 

tests,  17 
Promotion,  34,  88. 

Quartile,    19 

Rank,  percentile,  18,  68,  74  ff. 
Records,  school,  90 

Scatter,  19,  34 

Score    in    group    test,    40  ff., 

59  ff.,  71  ;  terminal,  63,  72 
Semi-interquartile  range,  19 
Simplex     Group     Intelligence 

Scale,  55  n. 
Spearman,  C,  29 
Standard  deviation,  iS  f. 
Standardization,  37,  91 
Stanford    Revision    of    Binet 

tests,    24  ff.,    43,    47,    66  f., 

76  ff.,   79,   80 
Subjective  factor,  17,  22 

Teachers,     estimates    made 

by,   22 
Temperament,   90 
Terman,  L.  M.,  26  «.,  66,  77  f. 
Terman  group  test,  43,  47  ff., 

71 
Terminal  age,  56,  58,  63,  72  ; 

score,  63,  72 
Thomson,  Godfrey,  21  «. 


Yonkera,l^l.Y. 
11   Sept.l9ki3 

Mr. Harold  jL..Leupp 
Librarian  bniv.of  Calif. 
Berkeley, Calif . 

Dear  Siri 

v«e  have  recently 
looked  over  our  copy  of 
Richardson's  iviethods  and 
Experiments  in  Mental  Tests 
and  find  there  is  no  page 
95  of  text  and, therefore, 
the  copy  that  you  have  con- 
tains all  of  the  index  there 
is. 

Yours  very  truly, 

WORLD  BOOK  GOMPaNY 
Ernest  Hesse 


EH;wiP 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OP  25  CENTS 

Emmm 


KEB    2    tM> 


* 


"^^ttrTTTH^rryr- 


LD  21-100m-8,'34 


Y.B  63630 


51924S 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


' : '" .  *  Ami 


